When Robert was acclaimed King, was this done on the assumption that Rhaenys and Aegon would have to be murdered? Might anyone other than Ned have thought about it and objected?

No, I don’t think that was the assumption. For one thing, when Robert was alive, Aerys II was king and it wasn’t entirely clear what the line of succession was, given Aerys’ vacillation between Rhaegar and Viserys. 

I’m sure the more “pragmatic” of the rebels – Roose Bolton, the late lord Frey, etc. – would have counselled ending the Targaryen line violently, but others would have probably gone for either keeping them imprisoned but in a means befitting their station or having them take vows as septons/septas and sending them to remote septries and motherhouses respectively.