It seems logical to pay off debts as soon as you can as they're expensive. However, in your example, if I use that $2,000 to pay down (a tiny bit of) my mortgage, I put myself in a dangerous position by not being able to come up with some extra cash in case of emergency. I hope most people with mortgages wouldn't do that.
It all depends on one's risk tolerances are and what other options for obtaining money there are (like being late or underpaying the next mortgage payment).
Keep in mind that taking on a mortgage itself is a risky move, as people found out in 2008.
BTW, even paying a few dollars extra a month on a 30 year mortgage can be surprisingly effective, since nearly all the early payments are interest, not principal.
