Will you change your bank when it refuses to work with Firefox? What if most other banks do the same?

This is how things are in Android now – online banking, online games and even subscription media services are mostly unavailable to those who would like to use non-official OS.

website authors will want to limit their own audience for the benefit of some company?

Many websites already refuse to work with anything not-chrome-based – so website authors often don't care.

Banks see that as 'security', so they are ok with 'losing' a small percentage of customers who want 'insecure' devices. In fact they would hardly lose anything, as their customers usually depend more on the bank, than the bank on any particular customer.

For media providers, that is another 'anti-piracy' measure (DRM) – they will also happily sacrifice Linux users, as insignificant fraction of users, probably less then 'actual pirates' on Windows or Mac. Netflix already won't stream in high quality to Firefox on Linux.

For online game providers this will be easy anti-cheat measure – they will also not care about that insignificant fraction of user.

Each of those service providers would loose maybe 5% of their user base (probably less… as most users would eventually accommodate), but the affected users would use major number of services they care about.