I understand that your reasoning works in the situation where you only have tabs opened from a single parent item. In that case, the attachment title indeed brings more information about the file opened in each tab.
But in the general case where you do the same for multiple items, the information "Snapshot" or "Supplementary Material" is meaningless without the information of the parent item. This is the general situation for me, as I keep multiple tabs opened from different papers when working on a topic. The idea to display the attachment title only works when you know which parent item it belongs to. Keeping this situation will actually encourage people to add information from the parent item inside the attachment title, which is discouraged from the perspective of the information displayed in the middle pane.

Once we add file-type icons to the tabs, I suppose we could start using the parent metadata for the first attachment of each type, since you'd at least be able to see that it was a snapshot.

I think it will probably be better to keep a more consistent behaviour for all secondary attachments.
When many tabs are opened, the most important information to display is the information of the parent item, for all child items. Then, the second most important information to know is whether it is the primary attachement or a secondary attachement. This covers probably the large majority of the situations where you have one Primary attachement and one Supplementary PDF file. Knowing the exact attachement title is probably important if you have more secondary attachments, but I feel that this is not so common (considering that the Zotero connector is rarely able to download Supplementary Files).

I would suggest:
1) Keep the same tab name for all child items.
2) Display the tab title differently for primary or secondary attachements. For example, a yellow background on the file-type icon (or on the full tab title) would be sufficient to carry this information clearly.
3) Add the attachement Title as a second line below the full tab title when hovering over the tab title. In that way, you keep a fairly fast access to that information, without being forced to switch tabs, which can be very slow for large files.
4) Display the list of attachements within the same parent item at the bottom of the Info tab, maybe in an attachements sub-pane similar to the Tab Selector (see previous post on this). If people don't like it, they should be able to disable it from the View -> Layout menu. In this way, even if you select the wrong tab within the secondary attachements, you can still quickly navigate to the correct one from that attachements list.

Even for child notes, which open in a separate window: the top of that window simply displays the Title of the note, which is just repeating the text written just below. The information on the Parent Item is displayed at the bottom, but it would save space and remove a useless repetition to simply put it at the top, treating note windows the same as the other viewer windows.

You know there's a Filename option for the tab title?

That option actually uses the attachment Title instead of the Filename for the tab title. Probably some bug related to that one?
Another issue is that it does not rename the titles of the tabs already opened, even after restarting Zotero. You need to select each tab title individually to force them to update.
Screen recording
Debug ID: D1672347414

Even if it was working, I don't see how that would be better. There is currently no rule for renaming secondary attachments. So that will not help to show the Parent Item information.

That option actually uses the attachment Title instead of the Filename for the tab title.

Ah, that's certainly a bug. Fixed for the next beta — thanks. Unrelated to the preview issue in the prefs, which will also be fixed.

Another issue is that it does not rename the titles of the tabs already opened, even after restarting Zotero. You need to select each tab title individually to force them to update.

That's a known issue, but note that it's only a case for unloaded tabs.

Even if it was working, I don't see how that would be better. There is currently no rule for renaming secondary attachments. So that will not help to show the Parent Item information.

I was assuming you rename secondary files manually based on the parent and some additional description. If not, it wouldn't help at the moment, though I did suggest elsewhere that we could make {{ attachmentTitle }} a renaming variable, and then automatic renaming could be used for supplementary materials (if we added a variable for testing whether the attachment is the primary attachment).

But for the next version we'll add the attachment title after the parent metadata for secondary attachments (when not using the "Filename" option) so that it's more usable in the default case. We'll also be adding tab icons once we have proper icons for EPUB/snapshot, and we could consider some difference in the icon at that point for secondary attachments, but that would only help in tabs anyway, not in windows, which can only have plain text.

There may be a misunderstanding here: the "useless repetition" I mentionned was refering to the Note Title, which is currently displayed twice when opened in a separate window, both in the window title and in the first line of the note.

OK, but that's certainly not useless repetition. The window title isn't just for the top of the window — it's what identifies the window in app/window switchers, the Window menu, etc. It needs to contain the note content so that you can get back to the right window. We can truncate at a shorter length and then add parent item info, but it makes no sense for it to show only the parent item metadata.

And we can probably also omit the parent item title for secondary attachments, as suggested in https://github.com/zotero/zotero/issues/2731#issuecomment-1201382522.