Why do old planes have problem starting?
Planes and cars are different. I’m stating the obvious there. But planes are more weight-sensitive than cars. So plane batteries are smaller. They’re just barely large enough to start the plane with little reserve over that. Planes also sit outside and unused for longer periods than many cars do. So the batteries have more time to self-discharge (aka lose charge).
Some cables used in planes were (for a time) aluminum cables. Lighter, but not as conductive as copper. Connections in cables and grounding get surface corrosion and don’t conduct very well. Planes sit longer and get more chance to get this bad connections.
Cars are started with the clutch in to separate the engine from the load. Planes have no clutch. If the engine turns, the prop turns and loads the engine.
So all of this makes starting more of a challenge. And, there’s the idea that airplane engine tech is roughly 20 years behind cars. The lower economic incentive combined with the FAA certification process (instituted for very good and valid reasons) has had the effect of limiting some technology applications in small airplane engines.
On the flip side, planes have an alternator to make current to recharge the battery and run the electronics. But they also have a magneto to make power for the spark plugs too. You can turn off the master switch in a plane with no problems. Well, your radios and lights will stop working. ATC might be concerned of course, but your engine will be fine and not notice a thing. I had an instructor do this in flight to make that lesson clear.