Selecting single element should be enough for actions requiring a selected measure

One could just as easily suggest selecting any single element should be enough. That is, clicking a single note instead of a measure rest. Otherwise it seems just as arbitrary (how come selecting this one rest works, but not other rests). That's already how it works for the actual menu / keyboard commands to insert measures & frames; the palette code could use similar smarts.

Not really, there are lots of differences between how palette items are applied versus how menu commands are implemented. Probably eventually it would be good to reimplement things to work more consistently, but for now, menu commands are generally well-tuned to the specific of the command, whereas the palette code is necessarily much more generic (same code applies to many different element types).

As Marc said, there are several things that give different results based up on if you use the menu or palette. One thing that comes to mind is rehearsal marks, the menu gives you a blank rehearsal mark the the palette gives you a smart rehearsal mark. I take advantage of this and use the one I want for the given situation. The Palette seems to have more built in function such as this rehearsal mark and the Change instrument text (the menu doesn't automatically update the instrument change). I've had situations where all I wanted to do was add an instrument change and make it invisible (so I can change the instrument name in part properties e.g.). With the menu there's no undoing necessary to get to that point.

There is no distinction in the words for them in the menu/palette. This is really off topic though. The original suggestion is to allow any item be selected in the measure to apply a measure element. This is already the case for using shortcuts for system breaks and page breaks. (I've never used the palette for a break or frame). I'm currently working on a Baroque piece and press enter to add the system break after I add my last Figured bass no matter which beat it's on. No need to select the measure. Why you can't select a staff text and insert a frame is a mystery to me considering that. Other things like jumps should work like that as well. Jump markers are always at a certain point in the measure so there should be no need to select an entire measure to apply a jump.

The point is, the palette uses the same code for elements with only a minimum of special casing for different element types. Every special case requires special case with likely duplication of code and more opportunity for things to get out of sync in the future as one of the copies gets updated but not the other. So it's not by any means a no brainer that each and each palette item would be handled by the same code as whatever dedicated menu item exists for it. it's something we have to make a conscious choice to make a special exception for, and each such decision has consequences.

That said, sure, we can indeed add yet more special cases for this, and I totally support that, I think it's worth it in this particular case. I'm just trying to set expectations - palettes use completely different code paths from menu commands, and there is also value in having consistency in how different palette items apply. It's a balance. So, again, this is a good suggestion, but expectations should be set accordingly.

BTW, the code to allow breaks to be added via keyboard shortcut with random elements selected is pretty new, I recall adding that only the past couple of years. And this is a good case in point of a difference that makes sense - it's nice to be able to hit Enter at just about any point during note input or editing to add a line break, but do we really need drag & drop to show a figured bass element as a valid drop target when dragging a break from the palette?

I doesn't make sense for the drag and drop feature to do anything but show the entire measure that a break will be applied to. If it shows something else as a drop point, that's fine it works. Specifically in the case of frames, a frame is always inserted before the current measure or appended to the end of the score. It should work as well as system and page breaks in this regard. Anything selected should always put a frame in the proper place.