How much of this is mitigated by donor/recipient compatibility (haplotypes? is that the right word?)? Is there no such thing as a purely compatible donor?
Yes identical twins are fully compatible. The first kidney transplant was done between identical twins and the the recipient lived with no anti-rejection medications for decades from what I recall.
A 100% compatible donor is a identical twin or clone.
Besides the MHC haplotypes which govern large parts of the immune response to ‘self’ there is the actual makeup of the rest of the genetic material which may be presented as antigens and then recognised as foreign
There have been instances of kidney transplants between identical twins to withdraw immunosuppressants after some time.