Posted as a comment by M. Teague
Your points are good but they are strategic advantages, not tactical. To continue, the Romans were full-time professional soldiers who carried an enormous variety of tools and gear to allow them to march far, fast, and fight almost anywhere. Barring dense forest and cliffs, on any sort of open ground and not having to deal with cavalry archers the Romans seemed invulnerable. And I wondered for the longest time why.
I knew that the Roman legion was much more flexible and maneuverable than phalanx, could accommodate men in various positions but that still doesn't account for trying to get past a head-on collision with a (very strong) pike formation. But it sort of came to me when a few months ago I learned that the Roman Gladius is actually only a half-decent stabbing weapon. It is relatively fast, but compared to other weaponry (as actually tested against other swords) it is poor in stabbing. But this was how the Romans were taught to fight. Why would the Romans equip their troops with a poor stabbing weapon and tell them to stab with it all the time? There are even curved swords not designed for thrusting that do better against the gladius. The gladius is a fat sword and it can only penetrate up to a specific point.
Then just recently I saw the documentary (view-able on Youtube) Conquest: Roman Weapons. Peter Woodward in conjunction with reenactors made an excellent examination. A longsword is a better all-around weapon but it required room to wield and the techniques are either tiresome and you fare better with a smaller shield. A spear was good for holding off infantry charges but while cheap is useless afterwards in the thick of a fight. A large ax requires both hands, and a smaller ax while a good tool, requires swinging. The gladius is small and allows a lot of quick thrusts. A falcata or heavy sword was great at chopping and cutting but that is not only tiresome, it is slow compared to rapid short thrusts.
But beyond this was a technique for battle I never considered. The Roman legion was a highly offensive force. Contrary to what I believed (a counter-attack force that received the enemy) they push and drive into an enemy with that large scutum shield, preventing enemies from properly mounting their large attack. It as a weapon drove enemies into the ground and pushed them back into enemy lines. Romans were not tight-fitted in fighting formation, but had a little space 3 feet from man-to-man) between each other which never made sense to me until the demonstration as why.
AND THE BIG conclusion was this: the Romans at the front would use their large curved shields to hold off the initial bunching of a spear formation. Lots of people don't realize the scutum uses a horizontal grip, so when your arm is hanging loosely at your side, you're holding the scutum. This means you didn't have to hold up your arm across your chest or stomach to support that large shield; your shoulder did it for you. Two scutum close together curving around you could hold off 10 thrusting spears. The rank interwoven from behind moved forward, while the spears are planted in or busy working the front Romans, allowing the second rearward to step into gaps (remember, 2 scutum Romans are holding off 10 spears) and then use their gladius to hack apart the spears or drive them out of the way. No more phalanx. If the Romans had a dedicated thrusting weapon, this tactic would be impossible.
The Roman legion could fight the phalanx head-on and decimate it. Playing straight into the phalanx preference of fighting and force it into buckling destruction. The phalanx is a tightly packed bunch of men in which, almost irregardless of your weapon (you can pick any one of 100 around the world in any combination) you were destined to loose. The tight formation is used against itself and loses all advantages from the first advance.