Is it at all realistic for me to play 4 characters at once?
It is realistic for a single person to play multuple characters at once, especially with enough preparation.
Sentient magical weapons, while not a D&D player character (usually?) are another type of literary character in a plot --having motivations, flaws, specialities, and weaknesses. The level of preparation it takes to play another character grows with the number, quality, and intensity with which ot is portrayed.
Can I expect to be able to find a DM willing to do this for me?
I can't factually answer the question; however, some people will try anything once. If you cannot find a DM, locally, at a game store, online forum, etc, I can recommend the Mythic GM Emulator as a system for solo play. You could be your own gm and have ALL the characters, an entire UNIVERSE, no, a MULTIVERSE, AN OMNIVERSE OF CHARACTERS. YOU COULD GO MAD WITH CREATIVE POWER!
I believe it will be fun, but is this indicative of a part of DnD I enjoy most, that might translate better into another activity?
Yes. This is indicative of wanting to be a gm, an author, a game director, or possibly a playright or play-actor. This is personal experience. I make characters for fun. I don't aim to min-max, but rather create sub-optimal characters who are invested in a goal. I also like to write stories or come up with twists on classical character tropes that may be interesting to see interact with others.
I say this also because a GM has to make an entire cast of characters, which never 'feels' bad for me, but is actually rather enjoyable. You can make as many characters as you want, but there are obviously other concerns you must attend to if you want to have players continue and enjoy your game.
If you arent looking to start GM experience, you may want to investigate "Troupe style play". A quick search on google pulled up references to a game called Ars Magica which may have more advice for you.
Weapons as characters: The adventure zone podcast characters Duck Newton - a park ranger and Beacon, his talking magical sword.
[...]Find a DM willing[...]? Sourceless, personal outlook.
[...]Indicative [of another activity]? Self.
[posted on mobile, will edit later for appearance, or you can edit, if you would like!]