I don't see much for them to gain and plenty to lose if they let google build a big customer base.
I'm surprised Nvidia didn't attempt to partner with them rather than developing GeForce Now as a separate product, they could have taken advantage of Valve's developer relationship/client/store front and handled the actual running of the hardware, which Valve seems much less interested in.
Stadia on the other hand is just a demonstration of the level of hubris Google is capable of engaging in. They thought they could launch a brand new store with the twist that people would need to pay a subscription to continue playing the games they had bought and without the option of using your own hardware should you end up purchasing a capable machine.
VIVE is by HTC supporting SteamVR, but Valve has their own first-pary VR headset + controllers with the Index.
Steamlink has hardware is dead because it was bad. Steam controller is dead as well.
Steamlink now is an android app that might work and might not work on your device.
Partnering with Google on stadia would generate a sizeable backlash.
Or can't, these days, considering how strangled the GPU market is.
I think Amazon, Google, and Microsoft will dominate cloud gaming eventually.
So far, I can confirm for a fact it works on HTC Vive, Valve Index, Oculus Rift S, and Oculus Quest 1/2 (in wired mode connected to PC, just like the other headsets, or using VR Desktop for wireless; I tried both and stuck to playing it with wireless).