Hey Clayton,
You are largely correct! However, there is a threading the needle scenario I will go over.
For conventional financing, borrowers must state whether part of the down payment is borrowed or if they are obligated on a note. Checking yes on either question is going to be a red flag for the underwriter. The underwriter will then want to include the seller carry in the borrowers DTI and will double check that the borrower is meeting minimum contributions. There is a world where the seller carry could be below the CLTV/TLTV cap for a loan program. However, these issues effectively nullify the benefit investors are looking for with a seller carry.
It is possible with DSCR loans, but you will want to read the loan docs very carefully! You are correct that most, if not all, hard money lenders will not tolerate a seller carry on the settlement statement at closing.
Here is how seller carries can be structured: The borrower brings their standard down payment to the closing table. Then, after closing the seller reimburses the buyer for the down payment out of the proceeds from the transaction. Some buyers will even use a gap funder for the down payment and repay the gap funder with the seller reimbursement to avoid going out of pocket. Depending on the legal language in your note, you will want to be careful what you use as the security instrument for the seller carry. In some cases the seller can record a second on the property after closing. If the buyers note doesn't allow for 2nd's, there is another work around. Instead of using a 2nd to secure the loan, the seller can be added to the LLC and the debt repayment terms added to the operating agreement after closing. You will want to avoid ownership percentages for the seller in excess of 25% as this can technically trigger the due on sale clause. Operating agreements are not filed with the secretary of state making this unlikely, but better safe than sorry. As a final option, I have seen the seller carries guaranteed with a personal note or filed as a second against another property the buyer owns.
Its convoluted and a legal minefield, but its possible. Feel free to message me with any questions.