THis is why I hang on to every transformer type charger that comes my way. The rectifier diodes are simple and cheap to replace, I have not seen any failed transformers yet. Broken wires and broken ammeters,failed clips and even failed line cords.Every one repairable, to pass on at a profit.

 

Seems like a good old fashioned no nonsense mains transformer, a couple of sturdy diodes and a current limiter might do the trick - with a little bit of control logic to periodically cut the charge, load the battery, and measure the voltage so as to avoid overcharge.

 

If you want to have a go at this switchmode psu, trace back pin 8 to the 350V supply and see what resistors are in series, otherwise just remove the bridge rectifier and output diodes , fet,and bin it..

 

I would check the inputs and outputs of the IC. For starters, pin 7 ought to be switching.

 

Seems like a good old fashioned no nonsense mains transformer, a couple of sturdy diodes and a current limiter might do the trick - with a little bit of control logic to periodically cut the charge, load the battery, and measure the voltage so as to avoid overcharge.

That would cost a lot more than the fully electronic computerized charger. Probably not even $5 worth of electronics at bulk pricing, and able to offer a lot of features as well.
I do not recall reading the specific complaint, nor seeing what that ribbon cable connected to. Almost all of the chargers that I have repaired had mechanical problems, except for the one small keep-charged unit that goes through all of the motions but does not deliver a charge. I saved it because it has a nice connection cable with good clips.
Even a cheap transformer costs more than a stuffed circuit board, and the shipping weight is much more as well.

 

Almost all 50/60Hz power transformer+rectifier type car battery chargers are a disaster. They really overcharge the battery, I think because mains voltage is not 115VAC but more like 123VAC. Some use SCR's and have regulation though. Then there is the surprise! selenium diodes which are further pain.
I use your basic LM317+temperature compensation for a charge maintainer and that works well enough. I wanted to make something better with equalize/float functions but didn't find a great way for a Arduino to control a SMPS TL431 section, so the voltage/current could be controlled.

 

The common non-regulated charger is mostly used to charge a battery that will not start a car in cold weather, at least in my part of the world. Or in warm weather, occasionally. So it is not just connected and forgotten, but rather applied for perhaps an hour or two, or maybe 15 minutes, then removed and the car started and driven. So in those uses overcharge is not an issue.
Selenium rectifiers have not been used in those chargers for over 20 years, and certainly they did have issues.
Voltage sampling SCR controlled designs have been available for at least that long as well.