The gap you're describing is real and it does kill deals when the timelines aren't matched at the point of assignment, so the instinct to solve it upstream in the purchase agreement is correct.
The assumption doing the most work in your current thinking is that one contract serves all buyer types equally. It doesn't. The purchase agreement you sign with the seller locks in the closing date, the inspection window if any, and any contingencies you've accepted. When you assign that contract, your buyer steps into those exact terms. If the agreement says ten days and your buyer needs eighteen, you're either renegotiating with the seller after the fact, asking for an extension that signals distress, or the deal dies.
The cleaner approach most operators use is to build in more runway than your fastest buyer needs, not so much that the seller balks, but enough to accommodate private money timelines. If your cash buyer closes in ten and your private lender buyer needs twenty-one, a twenty-one to twenty-five day close in the purchase agreement gives you room for either. The cash buyer can always close early. The private lender buyer can use the full window.
The piece you didn't mention is the inspection contingency. Your cash buyer wants none; your private lender buyer wants one. That's a harder reconciliation. If you write the inspection window into the purchase agreement to accommodate the lender buyer, and then your cash buyer is the one who takes it, you've given the seller a right to back out window that didn't need to exist. If you leave it out to please the cash buyer and the lender buyer ends up needing it, your buyer has a problem their lender may not let them waive. Some operators solve this by running two separate buyer conversations before locking the contract, confirming which buyer type is actually committed to this deal before they sign with the seller.
Confirm any assignment and extension provisions with your real estate attorney, since what's enforceable on extension requests varies by state.
Which buyer type tends to be committed earlier in your pipeline, the cash buyer or the private lender buyer?