Just to follow up on this and summarize:
One random day, happened to be a day when I worked in the office, the Teams bar below the "Type a message" bar disappeared, including the GIF button. In replacement my "Type a message" bar had an abbreviated set of three icons in the right side of it. (There are images of both in this thread.)
Note that all my coworkers have the same laptops with the same software, yet I was the only one in my team who had this issue.
With regard to the setting "Optional Connected Experiences" - not only did my enabling that not make a difference, but on a coworker's laptop whose GIF button did exist, they did NOT have that setting enabled. So although that is probably worth trying, it's clearly not relevant for everyone, and it's very unclear what that setting truly does.
Our help desk couldn't help other than suggesting rebooting (which I did - more than once - didn't fix it). Also tried explicitly signing out of Teams and signing back in (I made sure to uncheck the settings to keep Teams logged in), but that also didn't work for me.
Then on a day I worked from home (not sure if it was the next day or the day after), I saw that the full options bar reappeared below "Type a message", including the GIF button.
Bear in mind, I don't think the "working from home" played a role since I am connected to the work VPN from home, and also I had not rebooted or logged off my laptop - and I had not closed Teams. I hadn't restarted anything since my efforts in the office - despite that, somehow it magically reappeared without intervention.
So possibly the fix is "Be patient, live without it for a couple days, and it will magically reappear", because trying to take positive action to fix it doesn't seem to work for anyone. In the meantime, you can search for GIF images in your browser and copy/paste them into Teams chats, as a workaround.