What actually controls the timeline
Three things set the pace: how fast you can get the house looked at, how fast you accept a number, and the closing date you choose. None of those depend on a bank, because there isn't one in a genuine cash deal.
Compare that to a financed buyer, where a mortgage approval alone can take two to four weeks, and the whole deal can still fall through after that.
Why the range is two to three weeks, not two to three days
Even with no financing to wait on, a lawyer still needs time to run a title search and prepare closing documents — that's a real step, not a formality, and rushing it is how liens or easements get missed.
What can still slow it down
A title issue that needs sorting (an old lien, an estate not yet settled, a name that doesn't match records) is the most common delay, and it has nothing to do with the buyer being cash or financed. The fix is the same either way: find it early, not at the closing table.