No. No it doesn't. Allowing the government to use your service does not thereby make it part of the government.
(Nitpick: "free speech" is a general philosophical concept; we're talking about the 1st Amendment, which is a particular application of one of many views on that concept)
Bingo.
I'm assuming you're a developer. Have you ever wanted to leverage a piece of code, say GPL licensed, that does everything you want it to, checks off all of the development boxes, would make your life amazing.... but were forced to go with something else because of compliance? Same thing here. Just because Twitter does everything functionally they need, if it forces them into situations that would run afoul of their requirements, then they're forced to go a different avenue.
But twitter is working with the government in hosting the tweets and thus assumes responsibility to provide a public square, in which case it cannot ban people.
Otherwise government should be banned from using twitter because by picking a private company to put government messages on, the government official (in this case Trump) is picking that company's existing policy (what ever it may be) as a means of banning people, which the government isn't allowed to do.