When you remove the driver, you are “direct driving” the LED, which is why they are brighter than before; they are getting all that the battery can provide, only regulated by the resistance in the wires, & the battery itself. You will also run into excessive heat problems, particularly if you use good quality batteries.
As you have discovered, you can only Direct Drive from one 4.2v battery. Hooking up two batteries in series will be too much voltage for a single (XML) LED. This sounds like what has happened when you tried to repair the two-cell light driver.
Drivers that can handle two or more cells into one LED are referred to as ‘Buck’ drivers, & typically have the windings on them as you have noted.
This thread lists some high-power drivers for single XML’s.
Oh - welcome to BLF 