If the transformer overshoots (and i do expect some 20-50V peak from it), i see a problem if this is charged into a 25V electrolitic
What overshoot is the classical "core and coil" bulk transformer, mainly when large part of it's load disappear (e.g. the halogen lamp burn out).
The aluminum electrolytic rating stand for long term exposure (maximum tolerance with regular ripple on it, what may appear for longer than 1second), for occassional short time surges they allow way higher peak voltages (the 25V ones allow at least 40V overshoots with 1 second time limit - the component start to dissipate a lot of heat during that overvoltage event, cumulated time for such exposure over the lifetime in hours).
The electrolytic capacitors do exhibit a self recovery on an overvoltage breakdown (the damaged oxide layer is grown up back by the electrolysis), but during the recovery process dissipate heat, therefore the time of one event is limited.
And unlike in the film capacitors (where the damaged part is disconnected), the recovery in the electrolytics is a real 100% recovery of the dielectric oxide layer. But as the recovery generate heat and even gases, it put the rubber seal under larger stress, so with the recovery process happening too often, the seal would fail sooner.
Now how about 12V DC input ? Do they allow it, or they count on the capacitor having at least 17V in it ?
It is the real minimum: LED's are usually in series triplets, what mean about 9..10V drop on them. Additional 0.5..1V drop across the input diode bridge (two diodes in series). In some literature is depicted the use of two MOSFET's as synchronous rectifier instead of the bottom side diodes, what save voltage drop of one diode. But in real products I've seen only the diodes.
So the regulated 12V supply (so an AC/DC adapter,...) gives only 1V minimum headroom for decent current regulation, tolerances and wiring drops, what is really a minimum for decent operation.
Some cheepeese models use even regular 1N4002, what mean voltage drop in the 1.5..2V range, what mean the 12DC won't work correctly.
The switch element in the buck could be moreless neglected (given the other drops and their tolerances).
It is not suitable for 12V battery operation, as the "empty" voltage of 10V is way too low. Only the types with the synchronous rectifier would work reasonably (so you would have to replace the bottom two diodes by MOSFET's)
Operating at higher voltage is OK, all of them would work correctly in the range between 13..24V. It is actually what they are designed for...
So virtually any notebook supply (15..22V) with at least twice the power rating would work well, only the halogen capsules could not be used anymore.