Would this be beneficial for the network as the hash power is now randomly spread out?

The network itself would not really directly benefit, because the total hash rate remains the same, pools are just a secondary construction. Better distribution of who controls the hash rate, though, is good for the long term health of the network in terms of centralization (either in terms of more even distribution between existing pools, or more pools). If a lot of miners were part of a single large pool, the owner of that pool would be able to perform a 51% attack.

So no, randomly switching is pointless, but choosing a smaller pool to join and stick with is a good idea to help distribute the hashrate better.

Would this also benefit the miner as they now have an increased luck % on finding the next block with the random pool?

No, you have the same probability of finding the next block no matter what pool you are connected to, because the only thing involved in finding a valid block is the number of hashes you can try.