Hi Todd, I have Macs that can't be upgraded past High Sierra and they work fine. However, if I could upgrade them to a newer operating system I certainly would.
There are plenty of good reasons to upgrade, but one reason to hold off would be any dependence on old, 32-bit apps: 32-bit app compatibility with macOS High Sierra 10.13.4 and later. Since Apple announced the transition to 64-bit apps a very long time ago, that shouldn't be a surprise. If you have any 32-bit apps including 32-bit system extensions such as may be required for very old printers or other connected devices, they won't work with Catalina or later. macOS won't even load them. As a matter of fact that's why I have an old iMac running High Sierra — it's been relegated to act as a print server. That example is an obvious edge case that affects a grand total of perhaps five people on the planet.
Another reason not to upgrade would be an objection to paying Microsoft and Adobe confiscatory fees for the privilege of using newer versions of their bloatware. If you are using High Sierra versions of them they won't work with recent macOS versions. Fortunately there are plenty of alternatives available to both those legacy products.
Compelling reasons to upgrade include advances in Safari's iCloud integration, the ability to autofill two-factor authentication codes, automatic unlock with Apple Watch... things like that. Once you get used to little conveniences like that they can become indispensable.
Regarding precautions for Brave or Mozilla: I don't think any concern is warranted. Websites themselves need to be designed for whatever security precautions are appropriate for them, and if the browser you're using does not comply with them they will simply refuse to load — such as you are now experiencing with High Sierra's Safari version. Moreover, many websites implement their concept of "security" poorly, which leads to miseries when a banking website (for example) won't work with Safari. Unlike other browsers Safari takes a hard line on security and expects strict adherence to security certificates with no exceptions. Isn't that the way it ought to be?
Regarding Brave in particular, I have grown to prefer it over Safari these days for the simple reason that it incorporates native ad-blocking features for which Safari requires third party Safari Extensions. Brave also incorporates its own search engine, though you can use others if you so choose. In my opinion Apple needs to catch up to Brave in those features for Safari to remain viable. Apple needs their own search engine and native Safari content-blockers. At present though, Google pays Apple unspeakable amounts of money to implement neither one, and as long as they continue to do that nothing will change.
Browsers in general aren't long for this world anyway. There are too many fraudulent websites, fraud is out of control, and it's only getting worse. Dedicated apps are the only solution.