On a bathroom sink, can you use a non overflow drain in place of a overflow drain?

[Jmgillespey]

It started with relpacing an old leaky bathroom faucet with a new one from home depot. The new popup required removal of a plastic insert in the drain hole. After install it leaked through the bottom seal. As it stands now the one piece sink/countertop has a huge crack and a chunk out of the bottom of the drain hole. I think i can fix the crack&missing chunk with propox 20. I would prefer to use a new non overflow popup in the drain to avoid future leaks. Any advice would be helpful.

 

[Reach4]

The vents often let the bowls drain better.