Shuai chiao has always had a throw that looks like Tia otoshi as have many other styles of kung fu, for example that throw is one of the applications of the dai ma hung choy in praying mantis. In this video the same side grip being used is much more indicative of Beijing shuai chiao than the classic judo tai otoshi.

I also disagree that shuai chiao is strength oriented. It might be hard to be as subtle as some judo players are mainly due to the different grips being used but the throws are still based on leverage and using your opponents weakest angle against him.

It's true that there is a lot of cross training these days and most people have either done both, trained with some one who's done both or competed against some one who has done both. If your any good you're going to pick up on what your opponent is doing against you and you might like some of the techniques and try them. As someone who has done shuai chiao since the mid 70's and judo at a couple of different points, back in the early 90's for a couple of years and recently for the last couple of years I have seen both arts do things that I never used to see them do but saw the opposite art do quite frequently.