That's just... factually incorrect.
Armored vehicles (good ones anyway) don't stop moving unless it's a well-used MAV, and even then...
The way you used Falck was faulty pre-patch and is even more ineffective now. You can play your way; this does not mean the way you play(ed) is useful.
Like I said, you used to be able to join in the rocket spam, but that was it. An effective medic is one that's not dead. You shouldn't be charging in like that anyway; it's basically a free extra ticket for the other team.
Unless the downed player is impatient to respawn (for whatever reason; there are good ones), the downed-to-death clock is 30 seconds. Frankly, you'll get far more mileage out of waiting for the armour to move on or be distracted by another target than you will out of shooting the thing - that just tells the very-dangerous-vehicle-that-just-killed-your-guy that there are other squishy things nearby needing squishing.
Falck is by no means unnecessary. The nerf to Falck comes from the fact that her passive perk - the ability to revive to 100% - is now shared by two others. The syrette pistol is useful - it allows Falck to carry ammo, smoke or a claymore instead of a medpack - but in a firefight so many players are running around unpredictably that shooting them with the bloody thing can be a nightmare.