Lambda is backed by EC2 and absolutely has EC2's limits. Just like scaling an ELB takes minutes to serve "massive surges" -- AWS recommends you pre-warm ELBs or contact them to do so -- the same is true of Lambda.
AWS no longer recommends warming up the newer ALBs for the vast majority of customers and scenarios. Of course lambda isn't magic and has an initial start penalty. There are ways to mitigate it and as long as you are actively handling requests your functions will be reused. It's not magic, but it's capable of keeping up with normal traffic surges. If you have a special circumstance where you expect a large surge on a scheduled basis (say thousands of clients checking in at the stroke of midnight, you would have to arrange to be ready for that and there are a lot of ways to do it).
API gateway isn't alb or elb and there isn't any recommendation or expectation that customers warm anything up.
I never said "massive surge" but I stand by the idea that 2000 cores and 6 terabytes of memory is massive for most workloads. Of course, that's not the limit and you can request more if you need it.
Lambda isn't for all workloads but AWS (and everyone else) is always changing and improving. You have to check in occasionally to see if old assumptions (e.g. elb) still hold.