Earlier quoted context omitted.
JDBC is not JEE - it is core Java. Servlets are the original web-container Java spec, independent of JEE. Both are great specs supported by several stable and performant implementations - rock solid tech compared to a lot of the flaky stuff you find advertised today.
How are Java programmers implementing RESTFUL APIs these days?
Dropwizard and Micronaut are pretty good for getting a smaller and saner footprint. If you need more than REST, say you want a lean MVC-based web-framework, then Blade is also a good choice.
You can also choose not to use a framework and say use code-generation from a swagger/openAPI REST document and implement the remaining bits on your own. If your deployment target is AWS, you can also choose to leverage Netflix's fantastic set of libraries.
Java has tonnes of options, so you usually need to spend some time evaluating what goes into your stack.