Sending those thousands of requests is something your mail client does for you. Deleting 5000 emails takes a few minutes, but how often do you do that? I can select a bunch of emails in Thunderbird and just do stuff with it just fine.

For server side filters I just set them up in Fastmail using the web UI. That's the type of action I do once or twice a year, so totally OK to hop on over to the web app for just that.

I have no idea what you mean by 'push notifications'. I have Thunderbird open on my desktop, and it shows me when there is email. I have K9 on my smartphone, and it shows me when there is email (I don't have it set up to display notifications, but that seems possible). That's basically all I need to do email.