The moment you have a publicly facing API you are saying "these are the rules to talk to us". It doesn't matter what the client is, if it follows the rules then it should work.
A website is a publicly facing API and if two different browsers can talk the HTTP protocol and implement all the other APIs the website requires then you shouldn't be blocked from accessing the website through one of them
A website is a publicly facing API and if two different browsers can talk the HTTP protocol and implement all the other APIs the website requires then you shouldn't be blocked from accessing the website through one of them