Aside from the absurd prices of these drugs ($500/month) and the initial shortages (mostly caused by shortages of auto-injectors, not the drug itself); there doesn't appear to be many downsides.
They essentially reduce food addiction/cravings. Significantly reduce T2 diabetes risks via two distinct mechanisms. It is interesting the moral stance people place on this and how you don't see the same thing on nicotine patches or other cessation products.
Seemingly if people are overweight that is a moral failing, and doing something that helps live a healthy lifestyle is also a moral failing. These drugs literally cause people to crave vegetables, and that is bad because they aren't raw-dogging it.
Ultimately the only problem here is the criminally high prices. Hopefully other countries can get a better rate so more lives saved/chronic diseases mitigated.