Valve already knew that Nintendo is against emulation prior to sending the letter to Nintendo. The important datum here is that Nintendo has legal arguments why Dolphin is illegal software and that's presumably why Valve decided to drop Dolphin.
no Nintendo might have enough arguments to make a DACM take down request, but from the POV of many lawyers most likely not enough legal claims to have any chance at winning the dispute in court which would likely follow
this is IMHO why they don't do a claim at all outside of steam
But Nintendo is quite worried about emulation on Steam especially with the Steam Deck. They are obsessed with controlling the games through the hardware and have low end hardware many people would love to replace with something more beefy even if it's not officially supported. So I would be surprised if Nintendo hadn't pressured Valve into an agreement that they would reject and tell them about any Nintendo emulators long before Dolphin tried to go onto steam.
Now due to the lack of DMCA claim and lack of access to steam Dolphin has two choices:
1. give up on it (for now)
2. provoke a legal case in one of multiple ways which would legally clarify if they are covered under law or not (which is expensive and while not very likely there is always the risk to lose it). Through if they are unlucky winning doesn't solve the problem as Valve could just add "no nintendo emulators allowed" to their TOS in which case they would have to sue Valve and Nintendo together for unfair market practices.
... as a non profit project it's pretty clear which option Dolphin can practically take, especially given that you still can put it on a steam deck etc. just with a bit extra steps.
Whole lotta risk for 0 net gain, and possibly steep consequences.
If Nintendo had sound legal arguments, they would have already won a lawsuit, ergo it is reasonable to assume that they're just being Nintendo (i.e. loudly screaming that emulation is illegal regardless of the actual law).