Interesting question! I agree that the best option is to run the logistic regressions and then the other continuous models seperately. Tokens that can't be measured shouldn't be included in the continuous models at all, imo.

Regarding your last point, I just think that care is needed when contextualizing your statistical models in the results/discussion. Remind readers about base rates of glottalization across tokens as well as how the realization compares across tokens when it does happen.