I'm with Stephen and Jax on this one. I think it's logical that the borrowing club has the ability to heal the player they have on loan.
People are trying to equate the loan system in RS to real life. The trouble is, broadly speaking, here we're using opposite to reality in what is happening with regard to training and healing. As in many things to do with RS, real life scenarios don't necessarily work and we need to move away from trying to make real life comparisons for everything.
In real life, the club with the loaned player train him, not his parent club, but the parent club will keep a watchful eye on that training to ensure he is developing how they want him to and progressing. If they aren't happy, will say something or in extreme circumstances cancel the loan.
With RS, and logically so, the parent club train the player because 1) more often than not, they have a higher level training centre than the borrowing club and 2) they spend the xp the player earns as they see fit (the actual training) so the player develops as they want him.
In real life, the borrowing club treat minor injuries, albeit with the same watchful eye of the parent club on them, but may return the player to his parent club in the event of a serious injury unless the parent club are happy for him to stay and be treated where he is. In RS the borrowing club cannot treat the player but in RS there can be no medical negligence, risking damaging a player by healing him incorrectly or misdiagnosing or mistreating his injury.
In real life, a player doesn't rack up experience like he does in RS, so once he's played 20 games, his club wouldn't ignore his injuries because there is no benefit to them of healing him, but in RS that can happen. Why would a manager spend money and credits healing a player who has nothing to gain by playing more matches.