The PCB is a mix of through hole components and SMD parts. Near the AC socket is what looks like a switching mode power supply (SMPS). The RFI filter is not missing: it was left out on purpose since no provision is made for its components! The SMPS continues after the transformer. So far no smart charging circuit was met.
Moving to the bottom layer of the PCB, SMD parts appear and three SOIC8 chips are amongst them. The marking is easy to read: 5056. A bit of googling brings up the XA5056 datasheet from Xinna Semi. This chip is rated for 1A max current and "Mystery B3AC" exploits it for all three chips.
I tried a recharge with the box open (don't try this at home!) and measured temperature on the three 5056 and the PCB. In a minute or so IC surface was measuring 70°C (158F), which is about +40°C above room temperature. No wonder the whole thing gets hot!
XA5056 chips are configured to push 1 Amp into cells, meaning 12W (4V * 1A * 3 cells) must be dissipated somewhere. That value is configured with R2 (ref. datasheet) of 1500 ohm. I am considering of increasing it to 2000 ohm, so that charge current decreases and overall temperature doesn't go too high.
By the way, this chip looks similar to big brands product(s), even the datasheet gives clues... but I haven't found yet the equivalent.