SHA-256 PoW will probably work until it doesn't (if bots choose to invest in ASICs, or services that offer this pop up). Also users may be at a disadvantage as JS crypto would not be optimized for PoW (for example lack parallel crypto capabilities or context switching between calls).

One advantage a PoW "CAPTCHA" system holds is that the service operator can change the algorithm whenever they want. This may make an ASIC approach too risky to bother with. The JS<>ASM crypto bridge would nevertheless require some optimization from the browser developers.

Some cryptocurrencies which aim for ASIC resistance create PoW algorithms that would require re-implementation of a significant fraction of the the CPU die to be a viable ASIC attack vector. An example of that would be randomx[1]. Using it for in-browser PoW would require native support as it will not be competitive against the bots with just a JS or WASM implementation. A modification would need to be made to not be abused for crypto mining. This will also link the cost of the PoW solution to the opportunity cost of mining the respective cryptocurrency which is well understood.

[1] <https://github.com/tevador/RandomX>