Having the halving property for each step means you can easily create booklets by getting paper one size larger and pinning it together in the middle. That’s pretty useful.

Sort of. For scaling something to different paper sizes, the constant aspect ratio is the important part. But the subdivision property is also important for a few reasons. Take booklet printing as an example. You need a paper size that's twice as wide as the normal paper size to print that on (so Ledger/Tabloid for booklet printing Letter pages). But ideally you'd want this larger paper size to have the same aspect ratio, so you could scale up something like a poster to it. The only aspect ratio that works for is 1:√2.

Same for printing two copies per page (2-up). With a 1:√2 ratio, you can perfectly fit two copies of something side-by-side on the same paper size. This was incredibly common back when I was at school, where A4 worksheets were printed 2-up on A4 paper so that each individual one was A5 in size (half the area, √2/2 the length). With A4, you then just chop the printed pages in half and the worksheet fits perfectly. With any other aspect ratio, either there'd be wasted space due to the different aspect ratio of the chopped-in-half paper to the original, or you'd have to print 4-up on larger paper and chop it into quarters. The 1:√2 aspect ratio of ISO paper sizes means you can just chop a page in half and get the same aspect ratio, and that's useful to people doing printing, not just manufacturers.