I agree that the threaded reading and search facilities of many mailing list web interfaces vary between bad and appalling. But the principal advantage of mailing lists over web fora, however good the forum, is that with a mailing list the choice of client is yours, whereas with a web forum you get whatever software the admin has decided to run, for better or for worse. I thnk this observation goes far in explaining the very strong preference for mailing lists (and consequent dislike of web fora) amongst a certain generation of sysadmins. The recommended workflow for doing what you want is to subscribe to the list, build your own archive, and then browse it using the mail client of your choice.
Sites like the dear, departed Gmane were very useful for keeping archives of important mailing lists open to the major search engines, so that questions with explicit answers (eg, "What's the best path to upgrade a NetApp filer cluster from 8.3.2P9 to 9.3P17?") could easily find those answers if they'd already been posted. But they weren't, as you point out, great for browsing.
Don't let me stop you searching for the perfect web tool to search your personal archive of any given mailing list, though such a search is off-topic here. But I think you'll find that a mail client with good threading capabilities - though I don't use it myself, people speak highly of mutt - is your best bet for doing what you want.
I stipulate that this answer ducks the question of how you get hold of the list archives, but you ducked it in your question by presuming you could get them to run your own web tool on.