Configuration generally kind of sucks to manage, you don't get some of the role inheritance you can get from standard EC2/IAM, and forget running inside a VPC.
The idea behind using something like API Gateway + a Lambda for the various end points sounds awesome in theory, but going beyond contrived examples shows that we're a long way out.
Cold start time is a known issue but frankly, not that big of a problem in practice for many apps. Definitely I feel that dismissing this out of hand (I even got a downvote) is more of an emotional reaction to something really different and new than a rational one. There's nothing contrived about implementing a REST service with FaaS -- it's just simpler. It's worth giving it a serious look even as an experienced Java developer (as I am). There are a lot of benefits to the architecture, like minimal devops and tremendous cost effectiveness.
Don't think I'm dissing Lambdas (or anything like them in a non AWS context). There's a lot to love about them, not the least of which is that for small (in time to completion, request volume, and memory usage) units of work they are significantly cheaper.
There's a config management nightmare for everything but there are known ways to answer it for a traditional server/fleet approach. Run time configuration of Lambdas is still evolving and there aren't many options to choose from that are much better than just deploying the code and config as one as part of a delivery pipeline. It's workable now, but there are gaps. There aren't any gaps with running your own applications, it's literally what's always been done. There are certainly pain points, but the problem is already well understood.
AWS Lambda now supports Java, so perhaps we could consider API Gateway, Cognito and DynamoDB/Aurora as one of the new "frameworks". Google has their own competing offering as well, e.g. Cloud Functions. Firebase deserves a special mention as a rapid development framework, although at the moment it only supports Node.js. Like many frameworks it has pros and cons, but in my opinion full-on websockets support OOTB is a pretty cool pro.. especially since Lambda/Cloud Functions/FaaS can't do websockets on their own.