You don’t need CSS to make a website look different due to different screen sizes.
In other words, plain HTML is responsive by default. You can improve things with CSS. But if your site is unusable on small screens, you’ve broken something with your “design”.
Flexbox is by no means necessary to make a responsive web page. Media queries have been supported by mobile browsers for over a decade as has the viewport meta tag. Between the two it should be straightforward to make most page designs responsive.
Old browsers (old IE especially) that don't handle those features aren't a problem because they'll just fall through to the screen style. Trying to use JavaScript to help old browsers is a worse solution as you're more likely to run into JavaScript incompatibility problems than CSS.
Flexbox is a CSS 3 feature. Support appears to have been implemented in FF and Chrome around 2012-2014 and still doesn't fully work in IE[1].
The fact that a site uses CSS and HTML instead of Javascript does not imply that it's easier for older browsers to display.
Javascript 1.0 was released in 1996.
[1] https://caniuse.com/flexbox