Strange. In my market, a search in CL reveals dozens of vehicles in your target price range, several with <100k miles.
I've found car searches to be VERY location specific. I'm driving a 15 year old Ford Ranger for my side gig and the axels are about to rust off, so I'm keeping an eye open for a replacement truck. Hoping for a decent F150 with high, but manageable miles. I've been looking for something for over a year. Most trucks that fit the bill come in around $8-9k, while NADA, KBB and the like value them at $3,500-4,000. Nationally that might be true, but I'd have to drive 2,000 miles away to get a truck at that price. Not exactly worth it.
I would move on to the next vehicle.
A tough choice overall. I had a chevy malibu years ago that went through 4 head gaskets. Each time I'd be hit up for $1,500 to $1,800. You look at it and say "geeze, the car's worth $6k, I can't not spend $1,500 to fix it." But in the end, repair after repair after repair got old. When the last head gasket blew, a few of the windows, the heat, the ac, and a few other things stopped working, and I had enough.
I'd mimic what others say here. If the rest of the car seems reliable and well maintained, replace the engine. If not so much, I'd move onto the next vehicle.