It sounds like the initial problem was that the red hardpoint markers weren't visible for OP. I thought at first he had the issue that I did, but even though my issue is different, I felt like it was similar enough to mention.
This issue takes place outside of "cosmetic mode", where you place weapons and functional parts. That's where a) sometimes it stops letting me put parts on until I do the workaround I mentioned above and b ) there's a very limited non-scrollable window to show all your functional parts. If you have too many to list there, you just can't see them.
HOWEVER! The designer is less buggy in the latest update! I can actually scroll to the bottom of the ship list (which is still cluttered with scores of wacky space station designs for some reason) to see my actual ships. This is the first time in months that this has worked. Huge thanks for this.
The designer still grabs a ship hull from a totally random ship set, not from my civ's ship set, which is bizarre to me. And can I also suggest that the ships that automatically have their primary functional part allocated -- colony ships, constructors, etc -- that the part at least shows up in the parts list? Maybe pin it somehow so it can't be removed from the ship, but at least indicate that it's there.