In my games I wouldn't let one character rest and have the others continue adventuring without them. Either the whole party rests together, or no-one is resting. The party sticks together, so no-one is left sitting around doing nothing.
To this end I prefer the rest variant "Epic Heroism" from the DMG. This shortens a short rest to 5 minutes, which means the default behaviour after combat is a short rest, which barely impacts any time pressure.
Increasing the number of short rests in a day does make features that recover on a short rest better, but I haven't had a problem with class balance playing like this.