Because latency matters when gaming in a way which doesn't matter with AI inference?

Plus cloud gaming is always limited in range of games, there are restrictions on how you can use the PC (like no modding and no swapping savegames in or out).

Yep those are exactly the same considerations. LLM providers will have inconsistent latency and throughput due to batching across many users, while training with cloud GPU servers can have inconsistent bandwidth and delay for uploading mass training data. LLM providers are always limited in how you can use them (often no LoRAs, finetuned models, prompt restrictions)