Mounting BGA package on PCB with force without reflow
The Ironwood sockets work like this. They use an elastomer interposer with embedded wires. Depending on the type they’re good for hundreds to thousands of mating cycles. They use a slightly modified version of the BGA footprint, and are compatible with soldering down the BGA later.
The other BGA socket type uses pogo spring pins. These are more common for production test boards.
That said, it’s not feasible to just push the BGA down onto the  board with a clamp. Something has to compensate for co-planarity as well as make a gas-tight contact. Not only that, but pushing the chip down with that much force will damage the solder balls, making it impossible to solder the chip later.