Does it make sense for a single driver to own 2 cars? Debating whether to trade in my old car when I buy another or keep it

I'm an owner of a 2000 Honda Civic EX with approximately 241,700 miles on the odometer. I've had the car since 2005 and have recently decided to upgrade to a newer vehicle and keep the Civic as an extra commuter/beater car for local travel. After taking the Civic to the shop due to a clunking noise coming from the front of the car while driving, it was found out that the tie rods are pretty much shot. The entire rack & pinion assembly will need to be replaced as well as it has been leaking for some time, but it's been a slow, manageable leak. Other issues found by the mechanic include worn, leaking front struts and a leaking radiator, which will need to be replaced sometime soon, but can probably wait a bit longer. However, the engine, clutch, transmission, air conditioning tires and brakes are all still in good shape.
My question is sort of a two part question: 1) is this car worth keeping given its condition? 2) Does it make sense to own a second vehicle when I'm the only person that will use both? I do frequent short trips where I live and I figure it might make sense to put most of that mileage on the older car. The estimated cost for the repairs is about $1300.
So am I crazy to keep this car? Or should I just let it go and take the $500 the dealer is offering? I guess a big part of me doesn't want to let it go since I've had it for so long and I've always figured I'd run the car to the ground before getting rid of it.
I appreciate any advice or insight. Thanks!

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