{Showing a text block only in a specific part or the score} is already possible.

Actually it is possible in two ways.

In this example let us say that the text block should show only in the score.

By The Way:

If you enter the text block from Scroll View, it will automatically be measure/staff attached.

This means that the text block will not show in all parts, but rather only in that particular staff.

Method 1)

Use the context menu to unlink the text block. On the computer screen Finale changes the display colour, to show that the text block is unlinked.

In the linked part(s), use the context menu to hide the text block. Finale makes the display colour fainter, to show that the text block is hidden. (but you can still see it on the computer screen, in faint colour)

Now the text block will only show in the score (on the screen in a changed display colour, but not in a faint colour).

Some users may not like that the hidden text block displays on the screen (in faint colour), but then there is Method 2).

Method 2)

In the score, in Page View, drag the text block to the right until it is outside the (part) page.

Now the text block can not be seen in the parts (if you in the part zoom out on the screen, you can see the text block to the right of the part page)

Still in the score, hold down the Override key, and drag the text block back onto the score page.

Now the text block is shown on the score page, but in the parts it is still outside the part page.

I didn't realize measure/staff attached text blocks aren't linked. That's helpful!

There is something I like to do on parts that still makes me wish for a more direct way of specifying on which part (or score) a text block appears. On my parts, at the top left corner of each page I like to add something like "Flute 1: Symphony No. 9 (1/10)", where the numbers refer to which page of the total number of pages. I think that helps people collect and distribute parts. I can add that using the drag-off-the-page method, but then I'll have whole collections of these things piling up on score and parts that all have to be dragged off the page (except for on the part to which it applies). It's possible to do, of course, but somewhat of a confusing hassle.

… On my parts, at the top left corner of each page I like to add something like "Flute 1: Symphony No. 9 (1/10)", where the numbers refer to which page of the total number of pages …

The general way to do so is to use text inserts:

Text menu > Inserts > …

For the example

Flute 1: Symphony No. 9 (1)

the inserts text block would be:

{Part/Score Name}: {Title} ({Page Number})

where the items

{Part/Score Name}

{Title}

{Page Number}

are text inserts.

The only problem I can see here, is the total number of pages (= 10 pages in the Flute Part).

Since the total number of pages probably is different in different parts, this number will require a customized solution, like separate text blocks.

There is a Total Pages insert that will change to reflect the length of each part.

Thank for the comment, J Adrian. That's helpful to know!