I apologize if this has been answered, I just skimmed the thread.
Title and Caption search are a feature of iOS that I use quite often, if its not working for someone its a bug/glitch/setting somewhere. Thats a good thing though because that means it can be fixed.
As long as the photo stays in Apples photo library across devices you
can maintain its 'title' and search from the photos app in iOS should be able to find it even though title IPTC metadata isn't shown in iOS's info shade. I mention that because sharing photos to a lot of sources will remove that data since it can be privacy sensitive.
This is an image of a picture taken with an iPhone, automatically uploaded to iCloud and sync'd to my Macs Photos app, title edited on the Mac, resync'd automatically back to the iPhone via iCloud. So basically I just edited a 'title' and iCloud did the rest...
MacOS screenshot, added "Test Pic" as the title.
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iOS screenshot a minute or two later and searching "Test.." (Test Pic popped up prior to me typing it all out).
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Its showing the correct photo using the title I added. Same thing when I add metadata in a photo editing program like GiMP or Photoshop and drop it back into Photos on the Mac. However things like Creator and Author aren't able to be searched via MacOS or iOS photo app. You can see them with 3rd party metadata viewers though.
All the "test" is metadata I entered, the description and title (not shown) can be successfully search in the Photos app.
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While its not your issue I would recommend using Description in MacOS because its Caption in iOS. That means you'd be able to see and edit that metadata on both ends from the Photos app info shade on iOS. Just a recommendation vs using the Title tag that you can't see in iOS...
Apple takes all the photos and puts them into a package file (Photo.Library). The photos are given a GUID (unique identifier) as a file name for associating metadata with. There is a few databases that are also associated with that GUID for ecosystem stuff such as facial recognization data, live photo data, edit data of the original photo (photos aren't stored here just the data to create the edit on the fly), data for managing optimized storage, etc etc
including standard metadata EXIF and IPTC.
Meanwhile Siri/Search (previously Spotlight on iOS) has a database of its own that is constantly indexing with new photo data so it can search for your request quickly.
There are a bunch of things that can be experimented with to try to find the issue. I think the first thing I would do is edit the caption of a photo in iOS. Give Siri a little while to index it and then try searching for that caption on the iPhone. This will just rule out whether Siri/Search is just blind and its not building its database or whatever.
You can verify the metadata is on iCloud.com by logging on there with the Mac and downloading a file with the title metadata to verify it. If its there then log on with the iPhone and download the same file to the Files app from iCloud.com in safari to bypass your Photo's app sync function. Then send it to your Photos app from Files and see if the metadata is there. If so try searching for it again.
If you establish its a sync'ing and/or database issue you can try turning off iCloud Photo Library on the iPhone. Download and delete all the photos. Turn it back on and let it re-sync. If you don't have enough space you could restore the iPhone so it removes and resyncs.