I happened to buy a house with one of these issues. Back then, people still do "sitddown" drawing up of P&S agreements, at which point I got my lawyer, the seller's lawyer, the seller and me in the same room. The seller explained he'll need to stay in the house for the week following the closing. His lawyer offered to write in he'll rent it from me for the prorata amount of the mortgage, and he'll pay the taxes, water and heat. This was followed by my lawyer's "NO NO NO".
My lawyer explained that that would technically make the seller a tenant, and I would have to go through an eviction if he did what you did, either the tenant hangs around forever, or his belongings. With a normal P&S agreement, its simply a buyer/seller relationship, and the seller is obligated to vacate by the closing date. In this case, when I do the walkthrough prior to closing, if his stuff is still there, then no closing. If we close, then what's there is mine as I bought it, and I can dispose of it any way I please.
Now, I have bought businesses and real property where the seller for whatever reason cannot move something out by closing date. I had this happen with window AC's. To accomodate this issue, a list is prepared for "what is NOT included in the sale". Thus, anything not excluded is by default then included if you left it behind. I also had attornies asking me to prepare such a list.
The seller explained that he needs the money from the closing to then close on the house he's moving too. It's a big long story, but the closings dragged on and on, so I agreed to let him stay. I was also told that to make sure sellers move out completely, and NOT tell me something like "even if you're not supposed to be there, the new owner has to evict you", buyers should insist on a $200/day penalty if the sellers don't move out. To make things friendly, I declined to demand the penalty, and later on I regretted it when issues came up, and the seller and I had to go to small claims.
But that is another long story altogether. Bottom line is the purchase and sale of a house does not create a landlord/tenant relationship, and if you stay, you are a trespassing squatter.
Just to add, I've since heard of penalties that my lawyer mentioned, and from what I understand it's due to the attitudes of some folks like the OP here. I would not have believed that someone would do such things, but here is someone like the OP who blatantly declares "he can stay even though he's not supposed to", so come and evict me.
Guess you live and learn.