Generally, while they are in one piece and not moved, they are fine. But trasnporting or disassembling this stuff is a risk. The tiny wooden pegs, plastic bits and agressive screws going straight into soft board can rip out chunks, snap off, etc. Good luck getting a replacement door, etc.
I don't doubt you've had trouble, and I don't know what their website or otherwise remote customer service is like, but to say that the service is just not available at all in the US appears incorrect.
Assuming there is a store nearby and even that the company is still in business. It's vendor lock-in.
Meanwhile, my solid wood sideboard could be repaired by any competent cabinet maker, without needing specialist hinges or plastic brackets. Not that it is liable to fall apart as easily as particleboard & veneer.
IKEA can be repaired by any competent cabinet maker. IKEA doesn't use anything fancy and weird, all their stuff is standard or a slightly variation of standard and could easily be replaced by a standard one. In fact most of the IKEA stuff is so popular it has become standard.
IKEA sells both particleboard & veneer for their cheap/budget friendly stuff and hard pine/other wood for more expensive stuff.
You can't have cheap and solid wood, it's just too expensive.
Go down to the local wood shop and buy all the wood you need to make a sideboard and tell me how much it will cost. Certainly will be leagues higher then buying a particleboard & veneer version of the same thing.
Pine is a softwood.
Any novice woodworker/joiner should be able to fix IKEA w/o any parts from IKEA even.
With some practice, West System epoxy, filler and fiber glass can easily repair chips and chunks. It helps that I'm not fastidious about aesthetics, so lacking the laminate on top after a repair suits me. The little bits are more challenging to source if Ikea has stopped manufacturing or supplying them. I'm hoping 3D metrology and printing can help with that, so waste from people tossing out old furniture for want of a small part nearly goes away.
YMMV.
I recently ordered a replacement door myself, just to make it look new again.
Coincidentally this appears to be changing as of yesterday - https://www.bbc.co.uk/news/business-55796429
What's left are shelves and the two wooden sides of the bookcase itself. The shelves can be ordered: https://www.ikea.com/us/en/p/billy-extra-shelf-white-1026529...
So I would rephrase your "no item numbers apart from connectors" to "everything can be replaced except for the two sides of the main body". And even those you can get by ordering a new cabinet without any doors or extras, since that's basically the sides + a number of shelfs.
This is clear because, as you say, you cannot buy the sides, or the unique bottom piece on its own. And the piece which is most likely to degrade is the one which has many connectors pushed into particle board potentially multiple times - the sides.
Maybe it was a poor example, but I think my point still stands. There are many other furniture products for which no spares are sold, or happen to be available to purchase for other reasons.
Of course in many cases is easier (faster/cheaper) to replace the entire 'thing', altogether.
It's not like we buy them as show pieces.
ABS edging is cheap (like really cheap), can get a cutting tool for 10 euro, and glue the edge with double sided tape (or contact adhesive - more messy thought)