What is the "right" thing to do when a website tells me to download Chrome?

You are only forced by yourself. I regularly find myself ridiculed, ostracised, blocked, etc.; but I never find myself forced: in our western societies at least, nobody (outside of cops during demonstrations, but that is an entirely different topic) forces you to do anything. Instead, you are coerced.

The thing with coercion is that it is always a tradeoff between a small immediate cost (that will either increase dramatically over time, or be compounded by another factor because of some non immediately obvious synergy) and a big immediate cost (that will however prevent the situation from devolving any further).

Since most people aren't intelligent enough to spontaneously interpolate the missing information, and understand the entirety of the situation they are enabling, they take the small immediate cost, only to find themselves fucked in the long run, act surprised_pikachu.gif, and argue that they could not have possibly seen it coming.

The fact is that this lack of intelligence, while partly stemming from an unfair, abusive system, is still mostly on them: intelligence isn't innate. Like strength, it is an ability that one develops and cultivates, and even though there are definitely genetic and environmental factors, they account for a small part of one's overall agency in this regard.

So, had they bothered to be honest with themselves and just checked that their comfort might very well have an unexpectedly disproportionate cost (especially after people like me would have warned them), they might have realised the importance of not giving in to the easiest "solution".

Case in point, in this instance, the website is either checking for the blink engine (which could be a technical requirement), the user agent (which would likely be borderline legal at best - at least in the EU, I dunno in other places), or just using JavaScript to probe the browser (beyond its engine). Depending on the situation, the course of action differs.

In the first case, further investigation is required (as to why the blink engine is technically necessary, and if it is legitimate. Probably isn't, modulo some additional development cost).

In the second case, spoofing the user agent is a good practice, even if only for privacy purposes. However, pursuing a legal action (on the grounds of antitrust, complying with legally required accessibility, etc) is probably also a good thing to look into, or at the very least trying to find other people who have the problem, to try and organise a more permanent solution (like resisting the coercion as a group rather than as an individual).

In the third case, the situation is most likely abusive, and while there might be some reasons for them to do this, I would definitely investigate the technical aspect and seek legal advice (same remarks as the previous point here).

Either way, compliance is an invitation for bullies to continue bullying, so refusing to comply, or being difficult (slow, requiring as much work as possible on their end, etc), and implementing malicious compliance might be your best course of action. Use a virtual machine with chromium if you have to. Try to find a way to exhaust their server resources (and make it plausibility deniable, such as having as many open and hanging connexions as possible, submitting files in a loop, use fuzzing and pentesting tools, etc. Be sure to read the doc, not everything is legal). Make your documents as big as possible. Try to find bugs to make them unreadable, render only on specific systems (windows 98, etc). In office documents, use macros that exhaust resources. Etc.

And as I said, find others, organise, unionise, and become hard to deal with. At the same time, report the issue as clearly as possible. If they fix it, make sure to stop the malicious compliance as immediately as possible.