Can I ask specifically why it isn't feasible? Thanks!
Because battery charging requires a current-limited supply. Even though a modern switch mode PSU
does have current limiting, their control ICs implement, on purpose, modes that prevent the supply from running in the current-limited mode. It's typically either a "hickup mode" or "latch mode", but either one prevents it from charging: the power supply provides the constant current for a few milliseconds, and then just errors out.
If they did use a control IC with disableable over-current error modes, it would do the trick easily, but this is almost never the case: even though allowing the continuous constant-current output mode is trivial for a control IC manufacturer, the demand for such controllers is low enough that these ICs are fairly rare. The ability to do constant current is a huge "value add" for a manufacturer, even if it comes for free.
This is sad, since the power stage electronics might be able to do the trick easily, but given they are built for a specific purpose, i.e., supplying constant voltage to a load with limited capacitance, they won't handle constant current loads (big capacitors, batteries, directly connected LEDs, big motors...) for more than a few milliseconds.