Is there really no way to update Safari without updating the whole operating system?

The only option you have is to stop using Safari and switch to a third party browser. They are kept up to date for a long time, and many OS versions back.

I would suggest Firefox or Brave.

Is there really no way to update Safari without updating the whole operating system?

The answer to that specific question is no; it is not necessary. Safari updates are offered when Apple updates Safari. However, those updates reach a limit for any given macOS version, and Safari updates for that macOS version eventually conclude. There is nothing new about that.

You can keep using your existing version of Adobe Photoshop if you wish, but websites eventually demand newer browser versions for reasons that often lack justification. Try Brave.

On that subject, you might want to consider alternatives to Adobe's bloatware also. Their desire to keep squeezing revenue from its users will never end.

That's what we did after shutting down our home business, and now both recently retired.

Affinity Designer replaced Illustrator.

The free version of VivaDesigner replaced InDesign

Master PDF Editor replaced Acrobat Pro

The free version of DaVinci Resolve replaced Premiere Pro and Media Encoder

WavePad replaced Audition

I hung on to Photoshop because after trying many possible replacements because a) I'm just too familiar with it after 30+ years, and b) I use a few plugins a lot that don't work in other titles.

Adobe could help itself by pricing the apps more fairly. After years of paying $55 a month (which was peanuts next to the business income), $10 for Photoshop/Lightroom only is tolerable. But two things really irk the user.

After letting Adobe know I was going to kill all but PS from my account, they suddenly offered the whole suite for $35 a month. Really!? Why wasn't that the price from the beginning!? That and the single price per app makes zero sense. All of the major apps are $20 or $21 per month. Even if you stick with $55 per month for everything, then why isn't a single app $5 per month? Why is Photoshop alone $21 per month when the PS/Lightroom bundle is $10?