Yes, make a cash offer. You can always finance it later if you like.
As long as the buyer gets the cash, they're good
No. If I'm a seller accepting a cash offer, I expect cash (via wire transfer, yes).
I do care if it's cash versus financed, because financing comes with conditions (appraisal, inspections, etc.) that cash often does not. If you want to switch at some point in the short escrow period, and didn't have financing lined up to begin with, then ou're suddenly expecting to extend out the escrow, add extra due diligence, etc. All when I had accepted your offer under the pretense of it being a cash offer.
That's called "not performing" on the contract, and better do it during your due diligence period so you have another reason to back out, or they can keep the EMD.
If you want financing, make the offer that way, don't try to switch to it later. That would piss me off as a seller, as I've likely taken your lower cash offer over a higher financed one.
In this situation, as a buyer, I often make two offers - a higher financed one*, and a lower cash one.
*By higher, I mean higher than the cash offer, not necessarily high, could still be significantly below their asking price.