you are right, at the closing table, it all cash.
Too often the "All Cash" deal has a proof of funds from "Liberty Investments, LLC' assuring the seller the funds are avaialble. It is more often a smoke screen for, "Third party approval required before funds are released." So. "All Cash' can be code for invester looking for assingment.
Then there's the FULL PRICE offer from a buyer packing paper from Wells Fargo state this buyer is approved. Then Wells Fargo's apppraisal ends of 30% short. The buyer comes up with more cash or the seller drops the price or another solution is used. Then, 4 days before closing, Wells Fago 'finds something in the buyer history (or the loan is outside their lending ratio) and the loan is denied.
As you see, both present risk.
That is why many choose to hire a professional who knows how to squeeze and determine the real buyers from the pretend buyers. Yep, that cash offer would get kicked to the curb. Same with Wells Fargo.
Now the all cash deal with a proof of funds from Brooklyn Progressive Credit Union would be quite a different story. This a seller WILL prefer.
Buy the way, there is only one way to buy a home with a suitcase full of money. Take care you don't end up IN the suitcase.