Are there any solutions other than flashing Gapps, to have Google accounts on CyanogenMod?
You might wish to take a look at the NOGAPPS project at XDA. It's not yet complete, but most parts seem to be in a working order already. NOGAPPS basically is an OpenSource replacement of most Google apps (including Playstore), often mapping to free alternatives (e.g. OpenStreetMap for maps).

Side-effect: More privacy, less tracking :) But keep in mind that some components are still in Beta (Maps, Playstore) or completely missing (still under development; Login Services, Services Framework), so it's not yet a full replacement of GApps. Still it might be sufficient for many cases.