Th cap is marked 103 M and across the switch. Not uncommon at all. You would have to replace this with a safety cap (XorY) forget which. It's would be a 10 000 pf capacitor or 0.1 uf, but you would need 250 VAC. The cap is used to protect the switch contacts. The "voltage across a capacitor cannot change instantaneously".
This guy
https://www.allelectronics.com/make-a-store/item/tc-100/100w-up/down-transformer/1.html is too big. I piked up a nice 120/240 xformer with a switch from a surplus outfit which I plan to use for a project
Then you have these toroidal ones.
https://www.mouser.com/ProductDetail/Triad-Magnetics/VPT230-110
You can get switches and/or power entry modules which have the 120/240 switches in them. Torroidal transformers are expected to have the primaries in series or parallel. Other types of transformers are wired in the 0-120-240 tap kind of positions.
Notes of caution:
1) It might be wise to measure the secondaries for future use.
2) Transformer windings can be paralleled if IDENTICAL. If they are placed in series, they can ADD or SUBRACT.
Suppose you had two 120:120 V identical transformers and you wanted to make a 120/240 xformer at 2x the current. You would initially connect the primaries to a source <= 120 V and measure the outputs to confirm they are working. Mark the primary side connected to hot. Now connect the secondaries in series. measure the series connected voltage. Re-arrange, if needed, so the voltage adds. Assume a polarity, the black probe of your meter then mark as (-)wwww(+)(-)WWWW(+)
wwwww and WWWW stand for windings. the () are the polarities that you discovered. One side needs to be annotated. So, mark the hot in, and the two (+) for the phase markings.
Usually toroidal transformers come with the mounting hardware which is usually a large rubber washer and a large round washer.
So, if the secondaries are equal and they are placed in series, you will get either 2x their rating or nearly zero. Connecting two identical windings in parallel not phased properly, you might get smoke.
Here is an example of a series/parallel switch.
https://www.mouser.com/ProductDetail/Schurter/00334503
I have a piece of audio equipment which there are two versions. From the front they are identical. The later model contains the 120/240 selector switch. Internally, they are vastly different.