When I had one several years ago, it did not come with any linked debit card. I'd pay daycare out of pocket, request an invoice from them, submit that as an expense to the FSA provider, and they would handled reimbursement as an electronic transfer out of the FSA into my linked checking account.

So it's not quite as easy as you hoped, but it's still a way to get a "discount" on your childcare expenses with only a little effort. You're putting $5k into the FSA, but your take-home pay would likely drop only $3500-$4000 (depending on tax brackets). Your FSA contributions will be untaxed for federal income tax, state (AFAIK any of them) income tax, and payroll tax. So your savings could be in the 20-30% range depending on your income level and state.

A few quirks:

• The FSA will only reimburse up to the amount of the expense or what you've contributed so far that year, whichever is lower. However, if the expense is high enough that it' doesn't get fully reimbursed, the remaining expense amount should be kept on record and used to pay you as new contributions occur. Ideally this won't require you to submit multiple requests for the same expense.
• You don't have to submit each expense as they're incurred. You can "roll up" your care expenses during the year until you've built up $5k of payments and submit that as one big request. With $10k / year expected, you'd likely hit that around July, get paid about $2500 (the amount contributed to that point), then get the rest reimbursed as new contributions flow from your paycheck.
• With a single dependent, the $5k FSA benefit will wipe out any "qualified expense" that you get to claim for the "dependent care credit" on your tax return. But unless your household income is pretty low (<$30k-ish), the tax break of using the FSA is larger than the credit you'd qualify for.
• You can only select an amount to contribute during enrollment, and that is locked in until your next enrollment period. Any of that $5k you can't claim is lost at the end of the plan year. With your expected expenses, this isn't likely to be a problem, but unexpected situations can crop up.