Of course they have a choice. Just don't do it. All you said are predictions of what may or may not happen in the future. The opposite could be true - the audience at large may get sick of AI tools being pushed on them and prefer the browser that doesn't. No one knows. But even if you are right, supporting an hypothetical API that extensions and websites may or may not use and pushing opt-out AI tooling in the browser itself are very different things.
Sure, these features may never catch on... but if they do, consider the risk to Firefox: an underdog with dwindling market share that is now years behind capabilities taken for granted in other browsers. On the other hand, if these features don't pan out, they could always be deprecated with little hit to marketshare.
Strategically I think Mozilla cannot take that risk, especially as it can get feature parity for relatively low cost by embracing open-source / open-weights models.
As an aside, a local on-device AI is greatly preferable from a privacy perspective, even though some harder tasks may need to be sent to hosted frontier models. I expect the industry to converge on a hybrid local/remote model, largely because it lets them offload inference to the users' device.
There's not much I could do about a hosted LLM, but at least for the local model it would be nice to have one from a company not reliant on monetizing my data.