Safari displays unsafe website warnings for trusted sites

If Safari doesn't work as expected on Mac - Apple Support

If a webpage or link won't open on Mac - Apple Support

Don't omit any steps. You may ultimately conclude those sites require a newer browser. For Safari that means a newer version of macOS, but if you can't do that you can probably get by with Brave or Firefox. Be advised that even they will eventually require a newer operating system than Ventura.

Brave Browser Download

Download Firefox — Free Web Browser

Same goes for Chrome but no one should be using it anyway. Uninstall it.

My objection to Chrome is that it cannot be installed without installing Google, which infests your Mac with a plethora of system-altering components specifically designed to constantly scour it for personal information that it uploads to Google, in its inexorable quest to assemble a digital model of you which is in turn sold. That's how Google became the behemoth that it is. That information can also be given by a court or similar agency in response to a subpoena, then used and retained for whatever purposes anyone or any thing might want, limited to the imagination.

That's not a browser, it's a virus, and Chrome is the Trojan that delivers it.

I find that it works efficiently and updates frequently; it proved a satisfactory alternative to Safari, even before my iOS became obsolete.

Those are favorable attributes, and I agree with them. Brave does all of that, but without all the Google baggage. As a Chromium-based browser, it is otherwise indistinguishable from Chrome. You even have access to the enormous Chrome Web Store, which enables you to install browser enhancements that Apple may prohibit on Safari to due to potential privacy concerns:

Brave Browser Download

It's good. Try it. Then, uninstall Chrome — a procedure for which Google used to provide comprehensive guidance, but they no longer do. Wonder why...

Drawing your attention to this minor point,

I would be interested to know your objections to Chrome. I find that it works efficiently ...

If "efficiently" means system resource efficiency, Chrome (and Brave, for that matter) do not use system resources nearly as efficiently as Safari. In practice that means shorter battery life, fans blowing... etc. In addition to Apple's legendary privacy-obsessive features, Safari is specifically optimized for portable devices with limited battery life and modest resources. Nothing is as efficient as Safari.

But it gets worse, much worse. Google's automatically-updating, information-harvesting background processes are the real burden on Macs. That can be difficult to correlate because they run asynchronously, and whether or not you are actively using the Chrome part of it. Quitting the Chrome app has no effect on them.

Brave doesn't do that. Firefox doesn't do that. Only Chrome.

Oh and one more thing:

I will need a new iMac which hopefully will solve all these issues.

Lots of people arrive at the premature conclusion they need a new Mac when they perceive it has become "slow". One common reason for that is Google, which only grows and becomes more demanding of fixed resources.

(There are other reasons, but Google is often the first thing people install on a new Mac.)

If they arrive at that conclusion, Apple certainly won't mind if you go out and buy a new Mac every few years. Google won't mind because their processes run even better on newer hardware, as they always will. The only one left out of that relationship is you. You don't have to play that game.

Old but still relevant: https://chromeisbad.com/

Privacy? You have none: https://contrachrome.com/