macamoose

http://youtu.be/34F_S-FDu78?t=24s
If you would like to count how long this player stays on fire for, you should find it was about 10 seconds.

If you then take your attention to this:
http://youtu.be/CDfpfpUN00w?t=3m3s

When the Pyro re-ignites the Engineer whilst he is being healed, Engineer stays on fire for about 5/6 seconds. Therefor, the Medi Gun heals and extinguishes the flames faster than a player burning without a Medic with them.

Not to mention that there is an Achievement for extinguishing teammates as the Medic:

Not only that, but on a lot points are also given out to Medics healing players on fire with stats saying that the player was extinguished on the screen.

But why can't you get the "Allies Extinguished" part for the Medi Gun? I hear you ask.

If you look on the TF2 Wiki for the Strange Part....

http://tf2wiki.net/wiki/Strange_Part

If you scroll down the page you will see a paragraph saying:

"There are some mismatches between weapon capabilities and strange part compatibility which are not bugs, strictly speaking. E.g. the Medi Gun can be used to extinguish team-mates, but it isn't possible to apply the Allies Extinguished strange part to a strange Medi Gun. In this case, it's probably because the Medi Gun takes a few seconds to extinguish burning teammates, but all the weapons that are compatible with the Allies Extinguished strange part work instantly; only the latter mechanism is tracked by the strange part."

Hopefully by the end of this you will realise that I am dedicated as fuck to prove my point, sad as fuck to go through this in order to prove it and that you now believe that the Medi Gun can extinguish teammates.