Is there any possible way to obtain Mew in Pokémon Ruby, Sapphire, or Emerald?

Whilst there is no legitimate way to obtain a new Mew in original generation 3 carts any more, you can use either glitches or cheats to warp to Faraway Island and catch it in Emerald. The glitch method makes use of the Pomeg Berry glitch and is rather long and involved; full instructions are in this wiki how article: https://www.wikihow.com/Catch-Mew-in-Pok%C3%A9mon-Emerald

In terms of cheats, you should be able to use either Action Replay or Gameshark to warp to Faraway Island. (You can also use the cheats to spawn a Mew, but as you say, you need to carefully set the "fateful encounter" flag to ensure it will obey you). You can also use cheats to spawn the Old Sea Map item and run the entire event, but it seems that requires a few extra cheat codes to work properly. According to this site the Gameshark codes you require to warp to Faraway Island are:

D8BAE4D9 4864DCE5 A86CDBA5 19BA49B3 8DEB234A 4C8DC5EC

Another site mentions some other Gameshark codes that might work:

Alright, i found some codes. These are for the gameshark. Not the ar.

I tested these on a rom version of emerald- i'll be damned, they work. You must have the master and anti-dma codes on for it to work. The old sea map also needs the faraway enabler on.

Master code: D8bae4d9 4864dce5 A86cdba5 19ba49b3

Anti-dma: B2809e31 3cef5320 1c7b3231 b494738c

Replaces item in pc slot 1 with eon ticket: 121f112f da7e52b4

Replaces item in pc slot 1 with mystic ticket: 1901ddd1 aad31ead

Replaces item in pc slot 1 with aurora ticket: 667fc137 c4c35c38

Replaces item in pc slot 1 with old sea map: 16053ea7 b5ac6644

Southern island event enabler: 0d6a02aa b44948bd

Navel rock island enabler: 4465d861 df900be4

Faraway island & birth island enabler: C0443e89 7b97fec2 0976a51e fe4031b6