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Yes, it is almost always cheaper to keep an old car running unless it has major body damage from a collision or rust. (Or course you can continue the repair argument that you can just fix it yourself. Most body work is a lot of time and little material)
However, it isn't just about what is the absolute cheapest. If you keep a decades old car running it is still a decades old car. It doesn't have the features - convenience or safety - of a new car. Wrenching on a car takes time - and cars often break at inconvenient times. The last thing I want to do after working a 50 hour week is spend hours (sometimes days) wrenching on a car instead of spending time doing fun stuff with family. The older the car - the more frequent the repairs and eventually you will be continually chasing failed wire harness issues and sensors through a mile of wire connecting dozens of computers.
The cause of battery failures depends on how and when they fail. An early battery failure is likely a single cell or some sort of wiring failure. If you have driven a car for decades and the battery is just at useful life after cycle after cycle then it is likely all the cells are shot. (Toyota Prius batteries are very reliable and the vast majority of failures are high mileage / high year vehicles.) At that point replacing cells is pretty pointless and you either need to replace the battery with a new one, replace all the cells with new cells, or build a new custom battery. Or just find a new car because once you have worn out a battery with hundreds of thousands of miles of use the rest of the car is likely pretty worn too.
I'm not personally a fan of car loans either and since my 2003 TDI I have just kept saving that original payment after the loan is paid off so that by the time I need a car I have the money for the car. However, right now I have two car loans - why? Because with inflation at 7% and a used car loan at 3% the real rate is negative. The bank is paying me to use their money so I'll keep mine invested.
I'm curious about the circumstances that led someone to pay $300 a month ($3600 a year) for insurance on a used truck outside of the state of Michigan. Of course paying monthly costs more - usually 10 - 20% more but that can't explain $300 a month. I'm in the Portland metro and insurance here is the most expensive I've seen in 4 states I've lived (outside of Michigan). My last auto insurance bill was $539 for 6 months on 3 cars. (2004 Astro, 2011 Acura Sportwagon, 2017 Chevy Bolt) That is full coverage with 500K single limit, $100 comprehensive deductible, and $250 collision deductible.
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