How to make the underline rule thicker in Ai?

Yes, Ton, one may do some other workarounds to overcome the basic workarounds.

One may also just realize that Illustrator is pretty weak when it comes to flexibly underline text.

Ton, in the past I often suggested some similar workarounds. One problem is that it only works with (unrotated) single line point type objects or single line area type objects. (Well, some further very wobbly workarounds may work sometimes with multiple line type objects).

Have you already found a way to apply it to separate words or strings in single or multiple point type or area type objects?

Quite a long time ago when Illustrator did not provide the Underline Text option at all, there had been acute discussions with trench warfares between supporters and detractors of that feature.

Often there were some plausible reasons to support it and some dogmatic reasons to dislike it.

To me, the real disappointment is that it once was introduced in an incredibly ungraceful manner without any adjustable settings (compare it to InDesign). And no improvements since then. Pretty weak.

To make it a little less ugly create a line behind the type.

In the Appearance panel, create a newFill and drag it below the Characters item.

Use Effect > Convert to Shape... to make it a Rectangle an move it down with Effect > Distort & Transform... > Transform.

Save it for later as a Graphic Style.

Forget about that function. It creates pure ugliness and the one thing it's good for is to simulate a link in a web design.