Why does it need to look "modern?" If it's not broken, why try fixing it?

At very least there are some modernizations you could do for text legibility. Screens are bigger now, and today lines of text that run the entire width of the page are unnecessarily difficult to read. Similarly, with more screen real-estate you could pump up that font-size and line spacing a bit.

I'm having none of these problems, it's more legible than all of the "modernized" websites I've seen. Reasonable serif font, pure black text, pure white background. More legible than any posts on here, with the choice of gray text for certain things.

Here's why I feel the lines should be shorter:

https://baymard.com/blog/line-length-readability

And the font bigger:

https://www.smashingmagazine.com/2011/10/16-pixels-body-copy...

Now for smashing magazine to change their font colors to #000, for legibility's sake.

The line length is the main problem i would say.

It needs a max-width somewhere. This website was not designed for 25' screens.

I've just tweaked the page with:

    body {
	    width: auto;
	    margin: 2em auto;
	    max-width: 40em;
	    font-size: large;
	    line-height: 1.4;
	    background-color: #fffff8;
    }

    * {
	    line-height: 1.4;
    }

Consumer perception, mainly. It depends on the target audience of the page. Modern consumers have come to correlate certain appearances with quality. It's not an awesome cultural development, but you're not likely to gain much traction by fighting against that momentum unless you already have an oversized influence on that readership's expectations.