At my last job we built an entire API on top of serverless. One of the things we had to figure out was this cold start time. If a user were to hit an endpoint for the first time, it would take 2x as long as it normally would at first. To combat this we wrote a "runWarm" function that kept the API alive at all times. Sure kind of defeats the purpose of serverless but hey, enterprise software.
Did you actually need to?
That's one of the things that always threw me with complaints about cold starts - how many apps/etc do I use daily where I interact, and there's a 10 second delay before something happens? The answer: quite a lot.
Yeah, we can do better. And in fact, with Serverless, -most users will experience better-. It's only when load is increasing that you see those delays, and then it's still only a delay. Not shed load.
The fact I can experience that delay easily in dev makes people think it's going to be a huge problem, but, A. In real use it probably isn't as common as it is in dev (since you have way more traffic) B. You can design to minimize it (different API endpoints can hit the same lambda and be routed to the right handler there, making it more likely to be hot), C. It forces you to test and plan for worst case from the beginning (rather than at the end where you've built something, and now have to load test it).
Not to say to use it all the time, of course; there are plenty of scenarios where the cost, the delay, etc, are non-starters. But there are also plenty of scenarios where an engineer's instinctual reaction would be "too slow", but in reality it's fine; your p95 is going to look great, and only your P99 is going to look bad (on that note, a slow API response accompanied with a spinner is a very different thing from a UX perspective than a slow page load with no indication of progress), and even then it's predictable when it happens, and it forces function scale out rather than tanking a service. Of course, it's often not obvious upfront which scenarios those would be until/unless you try it, and that's definitely a barrier.