if they did the 2nd part entirely within your PC, would it be ok privacy-wise, and would the consumers be ok with it?

I mean Chrome works exactly like that now so, yes?

Depends on how you define “okay”. Do people understand how it works, and want it to work that way? Absolutely not. Even if they did, would they do absolutely anything to change it? Also no. And that’s talking about software that has a dozen excellent and free alternatives.