It depends on your programming language and how many libraries you are importing. If you use Python you can get startup times less than 200ms. Also, if your app is active, most startups won't be cold (Lambda reuses the VM a bunch of times before recycling it if it's staying active). You can also add a health check API and just poke that once a minute to keep it warm if you want (at the cost of increased invocations).

But yes, if you're doing something that requires a lot of setup, then this pattern isn't for you. But it turns out most web apps are simple enough it can be done this way, especially if you're already using a pattern where the frontend is doing a lot of the heavy lifting and your API is mostly just marshaling data from the client to the database with some security and small logic in between.