As you noticed, “user indexes” are in fact alternate table of contents: entries are listed in page order and don’t merge. Also there is only a single alphabetical index. So, to achieve your goal, you must cheat a bit.
The Alphabetical Index allows you to index on a 3-level key. Therefore, if you don’t already use the feature, first key will be you index fragment name (the name of your “separate indexes”).
As an example, I used first keys Index, People and Works. These will appear as names of your various indexes with the entries listed alphabetically below it.
Achieveing a more “separate” layout is just a matter of playing with the index configuration and paragraph styles.
Index configuration
Since we’ll have "separate indexes with their own headings, I untick Title in Type tab, *Type & Title. This suppresses the global “Alphabetical Index” heading generation.
Paragraph styles
The alphabetical index is controlled by Index 1 to 3 paragraph styles.
Contents 1 (level-1) is modified to have its text centred, bigger font size, bold with spacing above and below. Optionally, I added a forced page break in the Text Flow tab so that the index “fragments” are clearly separated.
Contents 2 (level-2 where we find the entries) is customised to remove the left indent (it made sense when level 1 is used for a primary key, but no longer is this case). You can adjust it for other effects.
Limitation and improvements
Since this is a hack on the built-in index feature, you still have a single index. This means all your fragments will be listed one after another. You can’t generate separate parts e.g. to have text between them.
The fragments appear in alphabetical order of their first key. You can eventually make profit of this to reorder the parts by prefixing the first key with a variable number of spaces if you can’t choose names with the “correct” sequence. But be carful because this will slightly shift the centred fragment “heading” and it can become noticeable.
Here is an example: AskLOMultipleAlphaIndex.odt (12.7 KB)
Note that formatting of the cited “work” does not transfer to the index (as is the case with all cross-references).
PS: update your LO version. 7.3 is at least 5-6 release late. My current release under Fedora 43 is 25.8.3.2.