I think a big thing to keep in mind is that politically, Sun Shao doesn't want to get into a scrap with the Cai. He cares about dunking on the Bai. His former heir was the one who keeps dragging it into a Sun vs Cai battle, and that is likely to be one of the reasons she was removed as heir. (Besides loss of face from losing in the finals.)
It's too late for that, to be honest. Shenhua switching from Imperial supporter to the new Cai/Bai/Xuan alliance (with Zheng support. Half the purpose of the realignment is to resist centralization) stole a march on everyone else on the political playing field. It's also the type of move that is disastrous to back out of half-way through, so barring another realignment of the board, the Cai/Bai thing is pretty much going to go through.
So what are the Jin and Sun doing?
Jin are sending a lesser scion, so a relatively small expenditure which gets them a couple things. First is an excuse to properly show up at the next political meet and greet at the sect (good job on Shenhua turning the Emerald Seas into a political center by co-opting an imperial institution a "neutral grounds" that totally favor her, btw). Second, following from the first, lets them sound out disrupting the Xuan addition to the Bai/Cai thing (which is less central to their interests). I'm not saying this is their only move towards their goals (they should be doing a lot else in other places that Ling Qi does not get to see), only that putting a scion here helps them get eyes on the situation that blindsided them and potential ins to try and salvage the situation of their monopoly being broken.
The afterthought goal there is probably to counter Xuan Shi, in that it's not expected but maybe the scion can humiliate Xuan Shi in a few years if they're both still there, or at least use diplomacy to make everyone like the Jin scion more than Xuan Shi (the Jin read to me as exactly the types who buy into their own hype about being master traders with silver tongues when their real advantage is their monopoly).
The Sun's goals are more subversive, I think, in a lesser replay of his gathering of dissenters within Thousand Lakes. They want to pressure Cai in a deniable way, not to stop the Cai/Bai thing, but to loosen her hold enough on Emerald Seas that it doesn't matter as much. Sending scions to make friends with Emerald Seas scions, selective "native" raids popping up and hitting some families and not others. Connections, and trade deals, and organization for all those who are discontent with Cai, who despite appearances is on somewhat shaky political ground mid move between Imperial Supporter to Old clan supporter. If enough clans shift to bothering by the book and slow down type tactics backed by "Of course I'm loyal, but I'm also loyal to our beloved Empress and her laws. I must make sure everything is done properly, blah blah" type excuses that can't be crushed and purged the way other types of rebellion would be it makes it more difficult for Shenhua to govern and takes up more of her time, effort, and attention. If Shenhua can be forced to turn her attention towards internal matters, and the trade routes between the Bai and the Cai made unsafe, this heavily neuters the Bai/Cai alliance and potentially makes it unsustainable. To Sun, the idea outcome would be the alliance collapses (due to lack of benefit), the imperial throne asserts more authority in the aftermath in retribution, and the Bai shrink in on themselves even more.
Sun isn't going to get all of that, but he likely thinks even a partial success will help weaken the Cai/Bai connection, and he's not wrong. I think he also has fundamentally miscalculated when it comes to Shenhua. Their wildly different background might lead him to consider her a politically weak leader who has relied on the support of others, and even her latest play just elevates her to politically savvy and opportunistic leader who relies on others. And in some ways he's not wrong (the Cai as a family are even more fragile than the Sun. Sun Shao, who is intimately aware of how fragile his own clan is, is probably reading his own concerns but larger into Shenhua's position), but fundamentally it's a misread. Shenhua relies on herself and her order, and she uses others and weaves them into her order. The Emerald Seas is not going to have the fault lines it should have, and it's not going to respond the way one would expect it to. It's still a work in progress, and it does have fault lines, but Shenhua has been shaping it into something slightly off from the old Emerald Seas and the Imperial Standard.
I also think the border friction is the leading contender for the flash point that accidentally turns this from a political realignment to erode centralization into an outright RoTK war setting. That's most likely at least a thread away though, but Yrs's style of worldbuilding is the type to have put the pieces in place, even if the whole picture isn't readily visible to the reader, long before the dominos start falling.