You do not get punished for having "a lot of credit cards with no balance on them" per se, but you do get punished for not having a balance on ANY of your credit cards. You should let ONE of them post a balance of 1-9% of that card's CL.
Okay, let me try one more time.
I have $40k in credit. I am paying off my last $4k of credit card debt. I will still have $40k in available credit once I pay that off because I have no intention of closing my credit card accounts. Do you get punished for having a lot of credit cards with no balances on them?