Completely correct - you should have your page size at the trim size, the actual size you want them.
Providing pages done as spreads to a printer is a nightmare for them and probably won’t touch them.
All your pages when made in a PDF for printing should be a single page in the correct running order, page 1, 2, 3 until end - set up as facing pages (as per @HotButton).
For printing the pages need to be in a multiple of 4 - so your doc should be 4, 8,12, 16, 20, 24 pages etc.
If you need further help, you know where we are