Evidently, I have LOTS of ideas to try here. Thank you everyone.
The thing is that judo kind of makes you stronger, but not enough for judo itself, and you need supplementary strength building to ensure your judo muscles are strong enough for the rigours of training (let alone handling other strong competitors.) Judo is a lot less strength dependent than wrestling, due to the fact that we can pull laterally on the other guy's jacket to unbalance him, and to push on it to retain our own balance. But, this also means that we don't develop as much muscle by doing it, even though it's certainly a great mobility and cardio workout! (In wrestling, you literally pick people up all the time, as I understand it, so that's lifting weight right there!)
The S&S programme covers so many angles of strength! I can't imagine anything else achieving this in such an efficient way as S&S does. There is a lot of movement, strange positions of balance and challenges to balance, and cardio going on in S&S. The moves mimic perfectly certain important/useful judo moves, and support the strength for others. It's also this close resemblance that makes it hard to train both perfectly together. Some have suggested going down to lighter weights for S&S, but I am already getting very good mobility and cardio from judo itself, and it's the multi-directional STRENGTH building that is especially necessary to get from S&S.
In any case, as is always the case, the answers I got were not at all what I expected to get, so I have a lot to think over and try.