Remote attestation is the true enemy of your freedom. The power of the authoritarian corporatocracy to force you to use only the (entire) systems they control. It's worth reading https://www.gnu.org/philosophy/right-to-read.en.html again just to see how prescient Stallman was.
RA is a technology that has its fair use, and can be desired for other systems, like in Linux. With a pure RA system your services can decide to trust or not those devices on your network that can be compromised, and report to other devices that there is something suspicious.
As anything, this can be used properly to increase the security of your edge architecture, or wrongly to limit the users actions.
Let me put another example. With RA I should be able to authorize validated systems in my R&D VPN. If you are using your own laptop with the company certificate, and the verifier tag the systems as "unknown" or "unhealthy", it will not allow the access to the internal network, but sure you can still use your laptop for anything else. This, IMHO, is a fair use of this technology.
We’re not just talking about the freedom to run software on your own device here, we’re talking about interacting with outside systems. There is an important distinction in context.
As long as it adheres to basic web standards, I believe no, the bank should have no say in what browser you use to access their webpage.