Gmail covers almost everything I need. I still use Thunderbird to create temporary IMAP backups (so I have another backup besides Google Takeout) and also to transfer e-mails from multiple providers just by drag and dropping from their inboxes or the file manager. Or when I want to edit the header of an email so that Gmail doesn't group them up in a conversation, this is a very rare / edge case that allows me to still use the conversations feature.
Plus labels and filters allow me to automatically "tag" messages based on sender/domain/wildcards/other variables
I don't have a huge amount of emails but I manage 3 Gmail accounts using a single client (with inbox delegation) and also older Ymail and Hotmail accounts as POP3.
But searching seems like something that is definitely don't better as a web service with people like Google and their extremely sophisticated and proprietary searching and natural language analysis algorithms. That's a reason to use web services rather than local clients.
as for natural language processing, I'm not aware that Gmail does this for user search? maybe it does, but I haven't noticed more than some basic stemming. and given the choice (is not hidden), even most basic users prefer exact/stemmed keyword search over "cleverly" second-guessing returning noise results you didn't ask for.
also there's many non-proprietary search result sorting and filtering techniques that do a great (and predictable!) job, locally, without having to go all deeplearning about it.
remember that a lot of the sophisticated NLP analysis stuff Gmail does, is for the benefit of targeted advertising first and user experience second.
Finding an attachment someone sent last week that you didn't have time to look at until now.
Finding a long set of instructions you sent to someone two years ago so you can forward it to someone else.