I'd say the only real reasons any founder needs "accelerator"/any kind of VC firm is lack of money and/or domain-specific professional help/mentoring when encountering a problem that seems to be too hard to overcome by yourself.

So, ideal accelerator/VC would be one that would want to fund your (ad)venture without taking too much motivation off by having large ownership/control, and have connections/pool of domain-specific professional expertise. And wants to believe in you.