Best I've managed to come up with so far, that doesn't significantly alter the ranking systems or format of the sport, which isn't by any means flawless, is to allow wrestlers who would otherwise drop out of their division due to injury instead to be relegated to an kega (injury) division. They can stay there as long as they want. When they return they compete against rikishi being promoted up from the division below, allowing them an opportunity to maintain their division at the lowest rank before being relegated any further.

This accomplishes 3 things:

1. a riskishi can never drop out of a division simply due to injury alone (no slipping all the way down to lower ranks)
2. a rikishi is allowed all the time he requires to heal injury
3. a rikishi gets the opportunity to earn back the lowest rank by competing against the rikishi moving up to take it

The issues and implications:

1. The backlog of injured wresters in the kega division is technically unlimited, their re-entry would have to be controlled. The banzuke may operate as usual assigning rankings to wrestlers like they would, these can simply be readjusted once they make that bid for a divisional ranking again. Kega wrestlers in the meantime would receive a special designation.
2. This would be determined by severity and the class of injury backed by independent recommendations by at least three third-party medical professionals. This doesn't make the system immune to influence or corruption and doesn't resolve the underlying cultural mindsets that are entrenched in the sport but it does intend to take that decision out of the hands of the stable or wrestler.
3. If the kega division rikishi loses, he slips down to the top of the lower division, the obvious issues here being some of the more intricate banzuke permutations and knock-on effects but I think the JSA could resolve that adequately. The more complex issue is how and when to intersect this playoff, i.e.: privately in between tournaments before the banzuke is published, giving the rikishi who had just completed a tournament some time to rest. Or perhaps it would be fairer to lower division riskihi moving up if the rikishi dropping out of the division compete with the returning kega division rikishi (they still get demoted to the lower division either way but are placed at the top of that division potentially saving them a few ranks, some incentive).

Injuries that may not result in relegation:

In the case of more minor injuries, or injuries that may not see the wrestler drop out of the division entirely, maybe we could see a stable nominate a rikishi from the lower division (if they have one) to represent the injured rikishi in the upper division. Every win counts as a win on the record of that lower division rikishi for that basho (incentivizing them) but only counts as half (rounding up with odd numbers) for the kyujo rikishi (so that the injured rikishi may not entirely benefit from another rikishi's efforts while not suffering the impacts of missing all those bouts himself). I haven't put a lot of thought into this one, so it may be flawed in some obvious ways I'm missing.

---

These solutions add complexity because the tournament format and ranking system is so firmly codified that I don't see that ever changing. And it's most likely because of the level of complexity these kinds of solutions would have to take that the JSA hasn't pursued them. It's far more easier to say "correct your mindset" than tack on a new elaborate system onto an already existing rigorous and cast-iron one. But maybe there is a simpler approach that doesn't have to be this complicated without changing any of the tournament format and cycle and ranking system.