Oh dear... Get ready for face punches. Is a $5000 car really a beater?
Agreed. $5,000 buys you a nice car in good shape, provided you're looking at the right models. Look at any of the car recommendation posts on this site and stop looking at ludicrously expensive cars. Heck, if a Prius fits your needs, go for it, but not the trade-in one you've talked about! $8-9k buys you a *really nice* used Prius (which is a pretty fancy car, honestly, even at base-level trim), so I can't imagine why you think you need to finance almost $13k to get one...
Also, you're leaving out an important part of this conversation, which is the amount of cash involved in the trade-in process. Monthly payment costs are not the most important part of the process- they're relevant, but leave out a lot of the big picture. You really need to be looking at the total cost to you- the 'equity' in your current car (insomuch as anyone can consider it to be equity...)
One final observation:
$13,750 IS NOT A SMALL LOAN, UNLESS YOU'RE BUYING A HOUSE.