My left hand and wrist hated me when I tried a 5 string, but I sometimes needed notes below that E. I tried tuning to C standard, but with my preferred strings (Ernie Ball Cobalt Flats 45-65-85-105) things were too floppy, so I went with DGCF. I'd only run across a handful of people having one bass tuned to that, but I quickly switched on all but one bass that runs fatter strings for BEAD. It's rare I need anything below D except when practicing to a youtube version in the wrong key. Then I detune to C and live with the floppies for practice, or I grab my BEAD and cope as I would with a song that needed it live.
I actually tried ADGC on the fat stringed bass, but it was too floppy so I tuned it to BEAD. If I need to jump on an E standard bass it's no trouble since I used that for years. I adjust quickly enough between these tunings.
All my basses are (reverse) Thunderbirds (see my sig for deets) and the EBCF tension works out just fine.
If there's a downside, it's that if another musician looks at my left hand to check what I'm playing, they're probably going to be confused!
Also, one bass (Epi Vintage Pro) has the bridge a little off, and intonation in D has the D saddle smack up against the bridge's guard rail. All my other basses intonated to D quite happily.