Our main reason for relying on the full browser is to not be tied to a specific browser engine, and to be able to benefit from browser security updates really easily, but there are obviously some challenges to this approach.
I don't think you need the extra layer of isolation. Browser tabs are isolated pretty well if they are running different sites. Chrome's spectre mitigations will also help to isolate functions from one another https://security.googleblog.com/2018/07/mitigating-spectre-w...
Chrome seems to handle memory quite well in headless mode at any rate. The memory footprint isn't tiny, but it seems a fair bit smaller than the container based approaches used by other serverless systems.