How do modern versions of Safari(versions 13 and later) compare to Google Chrome and Firefox in terms of ability to play videos and load websites?

If you are experiencing trouble loading websites or playing video content, then something is wrong. Review If Safari on Mac doesn't open a webpage or isn't working as expected - Apple Support, but your troubles are almost certain to be caused by attempting to use old versions with new content.

Have newer versions of Safari(Safari 13, Safari 14 and Safari 15) ...

All those versions have been superseded, to say nothing of the earlier versions you mentioned. It would not be surprising if they were to have trouble with recent website content. Bear in mind content creators always use the latest development platforms, and more often than not, prerelease versions of them.

To update Safari, update macOS. To learn how to do that please read How to update the software on your Mac - Apple Support.

At your option look into Safari Technology Preview - Safari - Apple Developer.

It's interesting that you mention that the days of browsers are numbered. Out of curiosity, what do you think the future would look like as it relates to web browsers?

A lot like today actually 🙂 Why, for example, would anyone use a browser to watch a Netflix movie? The obvious answer is that Netflix doesn't have a macOS app, but if they eventually get around to making one as good as their iOS app that there won't be any reason to use Safari with it. Netflix has demonstrated an inability to cope with Safari anyway: Netflix will no longer play in Safari - Apple Community among many other complaints with it that have persisted for many years and many macOS versions.

You originally asked specifically about videos, but the same goes for many other services. If you want to shop for something on Amazon their iPhone app works well. McMaster-Carr's app is a model of excellence. If you want to search for local services from coffee shops to law firms, you can use the Yelp app. Etcetera.

There's another problem with browsers, and that's fraud. Fraudulent shopping sites are proliferating at an exponential rate. They're not as pathetically obvious as they used to be, and lots of people who I thought would know better have been deceived into gleefully paying with Venmo or Paypal, money they will never get back, for something they will never get. Besides shopping sites, new Internet scams appear every day. Hopefully people are becoming wise to them, but if Apple Support Communities is any measure, that's not happening quickly enough:

Removing "Search Marquis" / "Search Baron" / etc on your own - Apple Community

Phony "tech support" / "ransomware" popups and web pages - Apple Community

Apps don't have those problems.

So yes, the days of browsers are numbered. They are certain to exist in one form or another, but their usefulness is becoming quickly supplanted by apps.

I don't consider it worthwhile to compare old and outdated versions of anything when content creators don't use them. Nothing gets outdated faster than software. Firefox for example releases public builds practically every few days, and beta builds more often than that. I can only surmise Apple is more interested in stability than getting updates out the door that often. They don't want a hundred million users piling on with complaints if a Safari release breaks something.

If Safari doesn't meet your needs, use something else. There are plenty of options.

I think the days of browsers are numbered anyway. One day everything will be an app. No one will miss browsers, and soon after that few will remember what they were.