I haven't looked into the specifics of RESPA regarding this, but my sense is that if the lender has agreed to have an escrow account and is collecting money from the customer, then it's the lender's responsibility to pay the taxes/insurance in a timely manner. Not sure you can separate the collecting from the paying.
But from a more practical view, generally the reason a lender will escrow is to ensure the timely payment of bills whose nonpayment could endanger your collateral/lien. If they're struggling to pay their escrow to you, then might it not be likely that they'd also struggle to pay the taxes/insurance bills directly? Then you've just got another problem.
And if you're collecting money from them but not paying the escrow, I could very easily see a customer getting confused with that--and then if a bill isn't paid some months later, they might come back at you with "what do you mean you didn't pay it--you're collecting money from me every month, aren't you?"
Sounds like a can of worms to me, but I'm not an expert.