If you mount the plate on a spring, and measure the steady state, the spring extends, showing that there is a force on the plate. There is no violation of conservation of momentum.

Look, this is pointless. Static states can isolate the individual effects, and then they all get combined into a dynamic state in varying amounts, and it becomes horribly complicated. People insist on trying to produce and explain overly simplistic models, and others insist on misunderstanding them. I've actually physically done these experiments and I know that what I say is true.

I'm not going to reproduce all the nitpicking tiny details in this forum because it's hard, inappropriate, and people will continually try to pick holes in it. It's the Monty Hall problem all over, and I'm just too tired to care.

There's every chance that your understanding is right in the cases you're considering, but I can't be bothered finding out where our experimental models differ.

I wish I hadn't bothered.