I used a similar budget and build something like this: 7x RTX 3060 - 12 GB which results in 84GB Vram AMD Ryzen 5 - 5500GT with 32GB Ram All in a 19-inch rack with a nice cooling solution and a beefy power supply. My costs? 1300 Euro, but yeah, I sourced my parts on ebay / second hand. (Added some 3d printed parts into the mix: https://www.printables.com/model/1142963-inter-tech-and-gene... https://www.printables.com…
I know it would increase the idle power consumption, but have you considered a server platform instead of Ryzen to get more lanes?
Even so, you could probably get at least 4x for 4 cards without getting to crazy. 2 m.2 -> pcie adapters, the main GPU slot and the fairly common 4x wired secondary slot.
Splitting the main 16x GPU slot is possible but whenever I looked into this I kind of found the same thing you did. In addition to being a cabling/mounting nightmare the necessary hardware started to eat up enough total system cost that just ponying up for a 3090 started to make more sense.