Does inclusion in S&P 500 even happen fast enough after IPO to matter here, thought there was usually a wait period before index funds have to buy in?
No.
The SEC's obligation is to all market participants, not just index fund holders.
These days? All rules are maybe this or maybe that depending on political winds.
Frankly, no company should be allowed to IPO without showing steady profit. An exchange or index that promotes IPOs of unprofitable companies diminishes its reputation, and the reputation of all other stocks it indexes.
Anthropic is free to continue selling shares privately. They can setup a website and take cc payments if the care so much.
12 month waiting period plus 4+ quarters of profitable earnings before you qualify for inclusion.