If by "old" you mean "minimally styled" then there are plenty of sites from that era that were really extravagantly styled, since it was a new medium that many people were exploring. There were also plenty of sites with Java or Flash that were considerably more intrusive than sites today (not to mention the period of time between when someone realized you open as many popups as you wanted and when popup-blocker plugins appeared).
Also, this is probably me getting old, but https://ucjeps.berkeley.edu/ looks quite modern to me.
But yes, good designs are not flashy, e.g. I love the design of Astro Starlight ( https://starlight.astro.build/), a starter kit for documentation pages.
So I also took inspiration from "simple designs" for my personal site: https://bryanhogan.com/
Why?
We can have bare, simple sites while still making them accessible.
Generally speaking though, I do think trying to paint 90s websites as some sort of utopian ideal of function and design is purely an exercise in nostalgia and nothing else. It is entirely possible to make fast, responsive, accessible, well-designed rich websites today, all without writing a word of JavaScript (not that including JS by itself is bad or anything). Do not mistake anti-user functions like heavy weight analytics and user tracking libraries, or poorly optimized and ill-architected code bundles as the current "state of the art".
Web technologies call out to the minimalist in me, but I appear to be in the minority.
I rescued the domain after it was left to expire and did my best to honour the original design from 2000.
https://chroniclesofgeorge.com
I eventually added proper css, bolted on https, and updated the html to something a little more modern and standards-compliant, but the site is still hand-coded, and looks pretty much the same as it has for a quarter-century.
The site loads in less than a second, you can do anything intuitively with a single click, all pages have a lot of useful information with zero fluff or clickbait.
check out their directions page: https://sokogakuen.org/info.html
I guess it's this one https://solar.lowtechmagazine.com/
Also like the style of Japanese websites where they seem broken/don't expand to fit available screen but cool aesthetic still
Although I see someone has put a 1.5MB image at the top, whose intrinsic size is 2000 × 2588 px, but which was downsized to 320 × 400 px. That's not prioritizing function.
My last use case for it was selling a car and giving away some free stuff. Sadly, those have been replaced by fb marketplace.
Because I don't want to deal with formatting, I want to focus on data. Firefox and Safari formatting looked great.
It's fast to navigate and order parts from, works on every browser I've ever tried it in, and loads very fast because there's minimal unnecessary components to the entire site. I hope they never change it :)
I doubt it's currently maintained, but these esoteric sites are fun
https://www.vannattabros.com/dozer.html -- A detail page from the site, not well organized but so much great info about heavy equipment and logging.