No, you are not missing anything at all. Not all fonts have OpenType features, and those that do may not support this particular function, in whole or in part.
It's a bummer because if it did, it would have solved your problem: the feature, implemented correctly, would replace the "th" (and "st" and "nd") after a digit with glyphs that are already drawn to scale. That means InDesign would not need ro "manually" make the font smaller and move the digits up – which is why the underline is off.
Not behind my InDesign at the moment but you could try using "Horizontal Scale" and "Vertical Scale" to make the type smaller, and "Shift Baseline" to move them up a bit. This is slightly different than a standard Superscript and if you"re lucky it leaves the underline as it is.
I solved this by creating a character style variation from my previous "Underlined" style.
As the line would be based on an apex character, I shifted the baseline down of 5.20pt and gave it something like 0.6pt of thickness. After the right amount of cursing I found the right values.
It looks seemless now.
Thanks for your help!