Unfortunately I doubt that "recaptcha solver" can be built as browser extension. Most of advanced bots for parsing automation built on proprietary platforms like ZennoPoster and they basically heavily modify Firefox / Chromium.
Also latency of human-recognition service is quite high so while you wouldn't need to solve it you'll need to wait for number of seconds anyway.
Few years ago back when I created such bots I had similar idea .
For certain this can't be mainlined. And if we talk about extensions then at least in past extension code didnt have enough capacities to automatically bypass recaptchas.
This would require fake mouse pointer control and it's obviously not one of features that extension api expose.
Also latency of human-recognition service is quite high so while you wouldn't need to solve it you'll need to wait for number of seconds anyway.