I think it should be rethemed. Even if its ties to SOS are obscure and buried, the tie still exists and whether one knows about it or not shouldn't be an excuse to keep it. Logistics-wise, I think that also means removing instances of "Zip-a-Dee-Doo-Dah" audio throughout the parks, editing out instances of ride audio from some fireworks soundtracks, and changing Disneyland RR audio as it passes through SM. I'm not particularly emotionally invested in any of that IP (I historically don't ride SM), either.
On a side note, I think the portrayal of POCs is much more problematic in Jungle Cruise. That's a bit more overt than SM's association with SOS, but that's another thread for another day!
If you're going to shut down something because of 'obscure' and 'buried' ties, then why stop at Splash Mountain. Just go ahead and shut down Disney World, because Disney also has 'obscure' and 'buried' ties. The studio produced the movie after all.
There is absolutely nothing about the ride itself that warrants a re-theme.