does paying for a traditional day care for my child count towards the child and dependent care credit?

Yes---if you paid for day care so that you could work, you can enter the daycare expense for the childcare credit.

https://ttlc.intuit.com/turbotax-support/en-us/help-article/taxation/child-dependent-care-credit/L8h8A5Klg_US_en_US?uid=m5yifys1

https://www.irs.gov/help/ita/am-i-eligible-to-claim-the-child-and-dependent-care-credit

Federal>Deductions and Credits>You and Your Family>Child and Dependent Care Credit

You will not get the childcare credit until (unless) you enter income earned from working.   The credit does not work unless you enter your income first.   If you are filing a joint return you must show income for both spouses, or show that one or both of you was a student or disabled.   If you have self-employment income and show a loss you will not get the childcare credit.  You will not get the credit if you are filing married filing separately.