CTH power supply

Years ago, while living in the US, I built a CTH (Cavalli Tube Hybrid amp) - the 'original' (well, the BoM is from 2011) version. I ordered a power supply (24 VAC output, with a 120 V input) and this all worked beautifully. I then moved to the Netherlands and bought a new power supply (with 240 V in and variable out, including 24 VAC) that worked briefly, once, and then died. Rather than buy a new one - I have a spare power supply from a ceiling lamp that delivers 11.5 VAC with max 210 W. I opened it up and it has a normal-looking transformer with a double set of windings in parallel. I figured if I put these in series, I should get 23 VAC which might be just enough for the headphone amp. The power supply is an EagleRise EET210CK. Mine looks slightly different, with a N (neutral) and L (live) by the input.
What perplexed me is that when I measured the output voltage I measured exactly 0 V. After much headscratching I finally figured out the primary winding's two contacts are not N/L but rather, measured to ground, each has 120 V. The output, similarly, has a voltage on each side, so again no N/L. Each side has a set of secondary windings.
Question is: can I use a power supply like this for the CTH? What happens if I take the secondary windings, put them in series and connect them with the output? Do I now have a N/L, or do I end up with some bizarre construction with no true neutral side? If so, would that matter?

Happy to post pictures of the PCB of this thing.

(Yes, I know this is all dangerous and working with little knowledge makes it even more so. But I learn from these exercises.)