Here’s a script made by me and Sergey Osokin.
Rename it from .txt. to .jsx after downloading.

The script would process the whole document and do these things with clip groups found:
1. Convert all filled clipping masks into filled compound clipping masks — it would make them clickable and draggable
2. Move all filled compound clipping masks to the bottom of their groups — this would fix the problem with them being moved to the tops of their clip groups and prevent selecting other children
Compound clipping masks without fills are ignored, as well as unfilled clipping paths.

Upvote this related problem to have it fixed someday:
http://illustrator.uservoice.com/forums/601447/suggestions/48267722

Can Adobe just make this easy? We want to make a mask, we want it to have a color, we want to be able to select it wherever we click, like anything else in Illustrator.

There is a workaround, but I have no idea about the nature of it.
If the clipping path is a compound path (even if it has path within) — it CAN be dragged.

+1000000

Could be there a reason for this kind of inconvenience, because it's difficult to understand why? Please change it to follow simple logic, allow selecting it by fill/shape, not from the edge.

Yes I agree. I have been working at Illustrtor for 15 years. I also know Corel very well. I totally agree. Rich editing possibilities, but very non-intuitive work. Confusing UX. UX Illustrator is a stone on the leg. Adobe team what are you?

I have a rectangle clipping mask around a circle (doesn't really matter). When rectangle has no fill, I click it and nothing gets selected — expected, fine. But when I fill my clipping mask and try to select it by clicking the colored area — it does nothing. It works for everything, except for colored clipped masks :) You have to drag over the edge only. It doesn't matter if you clipping mask is over or under (in this case, btw). It should work.

+1

Incredible oversight. Adobe sucks!

That's the foundation of illustrator I think, and it still there although it is completely insane to have to click on the edge to select it.
And if you want to select the colour of the mask, it is even more trickier because you have to click exactly on the edge, which can take several seconds only to pick one colour !!
Please Adobe change that !

I got Windows 8 Max, checked it in AI CC 2017, AI CC 2018

When i make a clipping mask out of a figure by drawing in it (using Shift+D twice) it becomes unavailable to select it by clicking on clipping mask figure. even though it lays on top of all other objects. It becomes selectable only through clicking on outline of a clipping mask figure. Have this issue in 2017 and 2018 versions.