Usually a decision factor between more serverless, or more DevOps salaries.
Why's that? Serverless is just the generic name for CGI-like technologies, and CGI is exactly how classical web application were typically deployed historically, until Rails became such a large beast that it was too slow to continue using CGI, and thus running your application as a server to work around that problem in Rails pushed it to become the norm across the industry — at least until serverless became cool again.
Making your application the server is what is more complex with more moving parts. CGI was so much simpler, albeit with the performance tradeoff.
Perhaps certain implementations make things needlessly complex, but it is not clear why you think serverless must fundamentally be that way.
But no, I'd not put any API services/entrypoints on a lambda, ever. Maybe you could manufacture a scenario where like the API gets hit by one huge spike at a random time once per year, and you need to handle the scale immediately, and so it's much cheaper to do lambda than make EC2 available year-round for the one random event. But even then, you'd have to ensure all the API's dependencies can also scale, in which case if one of those is a different API server, then you may as well just put this API onto that server, and if one of them is a database, then the EC2 instance probably isn't going to be a large percentage of the cost anyway.
There are probably exceptions, but I can't think of a single case where doing this kind of thing in a lambda didn't cause problems at some point, whereas I can't really think of an instance where putting this kind of logic directly into my main app has caused any regrets.
It can make sense if you have very differing load, with few notable spikes or on an all in on managed services, where serverless things are event collectors from other services ("new file in object store" - trigger function to update some index)