They do. I often get the message "This model is currently experiencing high demand, please try again later"
They do drop requests though? At peak times I sometimes have 1-2 requests fail with Opus; no biggie, the harness just retries it. But that, in aggregate, sheds load.
Thats not what Openai and Anthropic do though
If you want to go accuse someone of denying model nerfing, go find someone doing it and argue with them.