Photos not scanning faces/people - suggestions for other software including online?

Yes, the photos are scanned only once to detect faces, until something changes, adjustments added or a system update that is changing the face detection algorithm. During the face detection Photos is trying to find faces in the photos and videos. We cannot correct the face detection, only mark additional faces that Photos has missed.

The face recognition is something else. During the face recognition the detected faces will be compared, and Photos will try to recognise the detected faces belonging to the same person. We can help Photos with the recognition by naming the detected faces of the people most important to us. Photos will only use the faces it detected itself to learn what a person looks like, not the faces circles we added. We should check, if Photos accidentally merged the faces of two people into one album, because then it cannot learn properly, what the person looks like. The face recognition can be reset. If we remove all named people from the People album, Photos will start over with the comparison of faces, but it will not scan again to detect more faces in the Photos. Only the comparison of the faces to find faces belonging to the same person will be repeated. And we will have to add again the names to the faces.

Keep typing names below the faces circles, whenever you come across a photo of a person you named already in other photos, with a faces circle showing "unnamed". Probably Photos has split the faces of that person across several albums, because it has not been confident enough to merge the separate albums as one person. If you are lucky, you will name many faces at once, if this face belongs to a group of faces recognised as the same person. By naming the faces that are remaining "unnamed" you can merge larger groups of faces. It helps to keep the faces circles showing, so you can easily see, which faces have been detected, but have missing names.

What is the status bar below the People album showing? Is it showing "0 photos of 20000 photos scanned", or is there no status line any longer?

If photos is still sowing a status bar below "people" it is still scanning for faces, but if the status bar is no longer shown, the scan is complete.

Photos is scanning each photo only once for faces, unless you add new adjustments to the photos.

But it may be confusing, that Photos is not showing the results of the scan immediately. It is keeping the recognised people hidden and initially adding only a few selected people to the People album, only a few people it has recognised as the same person in many photos. This is intentional, as Photos 4 and newer does not allow us to reject detected faces. The People album would be very cluttered, if we would find all the faces of strangers in the background or people irrelevant to us in the People album. If you want to see more people in the People album, you have to find a photo of the person you want to add, where Photos could detect a face, then name the face to add the person.

Until you have set up the People album with all people you want to name, I would enable the option "View > Show Faces Names". this way you will see circles around all faces recognised by Photos. Typing a name into the name field below the circle will add the person to the album. Can you see circles around the faces of people, when you view your photos enlarged? Or are there still many photos, where faces circles are missing - even on portrait photos with clearly visible faces?

What's your budget? Among the alternatives:

Adobe Lightroom. This is a cloud app, not dissimilar to Photos but far more powerful

Adobe Lightroom Classic: Closer to Apple's Aperture app, more of interest to raw shooters, really

Mylio: a very good consumer app.

There are others, of course, depending on your requirements.

Especially with cloud storage - you're kinda trapped there at some point.

You're not. If you use Adobe's Lightroom and stop paying the subscription you have an entire year to download your images from the app. With Classic, if you stop paying the subscription you can continue using the app as a manager, just cannot edit anymore. There's a lot of misinformation about these subscriptions out there so it's worth checking carefully.