There is certainly nothing wrong with wanting to practice any style of TKD, or any other martial art, with flexibility limitations. Glad you found something you like.
There's a massive debate on what exactly each of those types of TKD exactly are, but so far as I can tell, there's effectively three kinds of TKD: Modern TKD, ITF, and Original/Traditional.
Modern TKD, which is controlled by Kukkiwon and largely responsible for what you see in the Olympics, is going to want to see techniques performed to a standard, so they want to see head kicks at head level (or to the moon if you're that flexible). You'll be just fine practicing this, but you may not win as many competitions.
ITF I understand a bit less than the other two types, but I believe its requirements of technique standards are the same as WT, meaning they want to see a technique performed to their standard. Same as WT, there's still nothing wrong with practicing this with hip limitations.
Original/Traditional TKD is the style(s) that were defined before everything combined under the Kukkiwon and split from what became ITF. You'll see these listed/portrayed as their Kwan name ("Taekwondo Jidokwan") or sometimes just by their original name, like Moo Duk Kwan. Many of these are actually Tang Soo Do, so maybe look into TSD as well to see if you like it. Regardless, and speaking from experience in an original Jidokwan studio, what matters is effectiveness. Yes, a side kick to the head is what the pattern may dictate, but if you can only kick to the ribs, it'll still be quite effective, which is the point. This category of TKD may be what you mean by Combat TKD.
To forestall any Traditional TKD debates, this is only my viewpoint and opinion on these topics. If you're interested, look up and read into the Kwans of TKD. Wikipedia is a start, there's plenty of interesting books out there for deeper reading.
And, in case it applies, any of these styles (as well as pretty much any other martial art) can and will improve flexibility over time, even if just by a little bit. So enjoy!