Why is a spin change favourable in intersystem crossing?
There are two things going on here. You have conflated with "forbidden" with unfavorable. In many systems, the first excited triplet is lower in energy than the first excited singlet. Hence, the order of energies excited singlet > triplet > ground state singlet is correct and sensible. If your question is "why is this the order?" that should be a separate question.

The forbidden part of the transition has to do with the fact that, because of conservation of (electron spin) angular momentum, you can't excite directly from a singlet ground state to a triplet excited state with a photon. The forbiddenness is a statement about not being able to get there, not unfavorability. The triplet is still lower in energy and the molecule has to do some other shenanigans (often involving spin orbit coupling) to do the intersystem crossing into the lower energy triplet state. If your question is "what is the mechanism of intersystem crossing?", that should be a separate question.