you said - 'You disagree that certain practices utilized in karate may lead to the same skills? Like push hands, one steps, and free sparring? Or even solo kata?'
in the main yes I do disagree,
this is why wrestling works excersise sets in a different way, mainly range led and of course tactically, but there is an intent that is different as well from most (not all) karatedo.
sure a degree of skill and application is developed from karatedo practice in the grappling range, but it clearly is a striking led art and thus developed that way,
historically I beleive this to be different however these skills and drills simply were not transmitted in any Ryu I have seen, or heard of.
So to clarify, good karate (a rare thing) does develop some skills in the grappling range (no where near to the level grappling specific arts do however), however IMO this does not include tactical ground grappling which is strategically not karate.