Roman legionaries were utterly badass if up to snuff. Among the best infantry the world has ever known.

I would argue that they are "the" best infantry the world has ever known. Nobody else in history even came close to that kind of discipline or combat effectiveness. I own a set of lorica segmentata, and tried to walk an hour and a half in it durring summer. I felt like an oven, honestly, I can't imagine what it was like for the Syrian legions.

It's tricky to cross-compare armies that never fought one another and which never had common opponents. I'd argue that one or two of the armies of the 17th-19th centuries could manage similar quality (Frederick the Great's Prussians, say, and the Red Coats of Wellington's army), along with Swiss pikemen. For several centuries, it was considered an established fact that the only thing capable of beating Swiss pikemen was MORE Swiss pikemen - hell, when a French force beat a small Swiss one (paying with heavy casualties), it was considered impressive... despite how the French had enjoyed a numerical superiority of about fifteen to one.

I agree, but cross historical speculation is one of my favorite activities, at one point I even had a long, protracted argument about who would win, given equal numbers and terrain, between Napoleon Bonaparte and Alexander of Macedon. I would also agree that Wellington, Frederick and the Swiss were fantastic troops, the best of their respective points in history, but I would argue that nobody in history has reached the level of absolute killing power that the Roman legionary enjoyed. The legions under Trajan would have torn a bloody swathe through Swiss pikemen the same way they did when fighting the helenistic kingdoms, all of which used the modified Macedoian phalanx, a spear formation that would make a Swiss pike wall look like children playing with pointy sticks. Not to mention how no Medieval army could muster even a tenth as many troops as even a standard Roman field army. Even with the armies of Prussia and England in the Nineteenth century, the only ways in which they could defeat a numerically equal Roman army would be with guns, and only then because the Romans might rout from being attacked by weapons which they don't understand. Take that away from the 18th century Europeans, as well as the inevitable advantage in numbers, not even those armies could withstand the Roman legions. They would get off one volley into the Roman's faces as they advanced, the Romans would close ranks, charge with Gladius and pilum, then massacre the unarmoured, poorly armed ranks of infantry who might, at the best of times, have negligible hand to hand combat training, and it would, like every other time a proper Roman army was engaged in fair combat, be anihilated.

Well, armies are developed based on what's available. If you're going to assess how Romans fought against other armies while allowing the "test" to be conducted on Roman terms, of course the Romans are going to win.

First off - the Red coats. Rate of fire. They were able to get off several shots a minute - to the point they could fire three or more in the time it took an enemy to walk through their range. The Thin Red Line was designed to DEFEAT mass attack - the French consciously modelled themselves off the Roman army in a lot of ways.
And there were other forces which posed trouble for the Romans at the top of their game, too. (Teutoburger Wald and Carrhae come to mind)

Oh, and the Swiss pikes were emphatically NOT "much worse" than the Macedonian phalanx. They were different, because the technology was different, but the swiss pike blocks were highly manoeuvrable - and not burdened with a shield, either. (They were, in a sense, Roman in tactics and Greek in weaponry - and the Romans didn't beat the phalanx head on, they outmanoeuvred it. They'd have much more of a problem with the Swiss.)

But the true "tactical counter" to the Romans? As in, hard counter? The Hussites. A wagon laager is impossible to outflank, the short range gun fire makes a mockery of any armour, and Romans who got into the centre of the laager would do so as isolated units. A Legionary out of formation is a dead legionary.

It is especially notable that anti-Swiss tactics were two step. Step one was always to disorder the pikemen - and it often failed, and if it failed then the step two, sword-and-shield armed troops (who used their agility, which Romans would have had less of anyway) were useless. A pike unit like the Swiss one, see, does not have to operate on a single axis - unlike the Phalanx.

Alright, maybe a member of the Coldstream guard with fifteen years experience might get off three shots in a minute - in the heat of battle - with friends dying all around - half choking to death in gun-smoke - arm aching from the recoil - poorly fed - boiling hot in a wool coat - but the vast majority of Red Coats were raw recruits at the beginning of any given campaign, whereas Roman legionaries didn't put one toe into combat without at least six months training. Not unless times were dire, anyway. Not to mention that a legionary isn't bound by the rules of mass-volley warfare, and didn't have to walk in step toward the enemy. Any centurion worthy of the title would order his century to charge the sixty meters or so that a musket could fire accurately after, or even before the first volley. And yes, English tactics were built to defeat a massed attack, but that argument assumes that a Roman commander would be tactically inept enough to not match the length of the opponent's lines, which was standard practice even in the ancient world.

As a note on your two examples of Rome losing: Yes, the Roman's lost at the Teutoberg Wald, but only because they were A) Betrayed, B) Taken by surprise C) Attacked by almost equally strong infantry that had set an ambush, D) Attacked from a superior position, E) Attacked by superior numbers, F) Were caught in foreign territory in which their fighting style was useless, G) Were ineptly commanded, and H) Were harried for three days of bad weather, no food, no rest, and innumerable traps set by the Germans. Could the Red-Coats survive three whole days against those odds? I doubt it. On the subject of Carrhae, almost all of those same factors apply. Except, instead of stronger infantry ambushing them, it was heavy cavalry and horse-archers, neither of which were the Legions ever built to face. Of course, a competent commander could have won that battle, instead of a fat-cat beaurocrat like Crassus. That isn't even to mention that the legions still worked by the maniple system at that point, and lacked the heavy handed discipline and highly specialized weaponry and armor that later legions enjoyed. I specifically said in my comment that I was talking about Trajan era legionaries.

You mentioned the Swiss pike blocks as being highly maneuverable. That may be true in that they're more maneuverable than a phalanx, but that's not saying much. At the end of the day, the pike is an anti-cavalry weapon, and competent infantry, of which there was almost none in late-medieval Europe, would have no trouble disassembling one, given proper process. Were I a centurion, I would suggest these tactics:
1) Order the first line of the century to hunker down behind their shields and press against the pike-wall, forcing the pikemen to slow down or abandon their formation.
2) Have the rest of the formation hurl their pila into the massed ranks of spearmen, killing many and disrupting the formation considerably.
3) Order the first line to push the pikes along the line, against the pikemen. The Swiss would have two choices: raise their 18' spears into the air, or allow them to be forced on an axis along the line, leaving the regiment helpless.
4) Have the first line pull back, and the rest charge forward and engage the pikemen in close combat, at which point their inferior armor, weaponry, lack of a shield, lax training would tell, and the box would be routed or massacred.

The Hussite wagon laager was an impressive weapon for its time, but it had one fatal flaw: it was stationary. If the Romans caught up to one, they wouldn't spend the blood assaulting it, they would surround it, build a wall of their own, bombard it out of gunshot range with siege weaponry, and wait for the occupants to starve. Roman's carried their rations with them, and any adept commander would keep resupply options open, either from a train or local foraging. Give it a week, two maybe. They'll surrender. Just look at the battle of Alesia, if the Romans have shovels, time and dirt, you've already lost. Also, I think you underestimate the effectiveness of a legionary out of formation. Yes, they're vulnerable to flanking, but so is every other infantryman ever. Being separated from one's unit simply demoted a legionary from "Almost unbeatable" to "Still really, really hard to kill."

You also forget the difference between the shields of the time of Swiss pikemen, the sergeant shield, a triangular shaped piece of plywood, and the buckler, a metal dinner plate with a strap on the back, were almost useless against pikes, whereas the Scutum was built to withstand heavy punishment. When I had to replace my own Scutum, I decided to smash it to bits with an axe. It took me an hour, and every swing left me open to a stab from a gladius. Every hack that didn't bounce off got my weapon caught in the leather. Sure, it can't stop bullets, but it can stop a hundred bayonet thrusts. The Romans might take initial casualties, maybe ten or fifteen percent, given the short range and awful accuracy of the smoothbore musket, but, once they reached the actual line of musketeers, and the British or Prussians needed to switch to swords or bayonets, the battle would end. I would be surprised if one man in a century died of bayonet wounds.