Just took the opportunity to try Thunderbird again and had to downvote myself :) I guess my previous frustrations were related to initial sluggishness while downloading and non-compacting loads tens of thousands of mails from gmail. It still takes some time, but after downloading and compacting my data, search seems to be very responsive.
Technically Grav is not a static site generator, it is just a flat file cms. It means there is no need to generate all the files of website and upload them to server each time you write a post. I have no idea why people like static sites for blogging.