Welcome to the Board!
I recently finished cleaning up the ZZ machine in the picture, and was having stitch problems although a little different from yours. Someone prior to me had removed the needle stop screw in the needle bar (I think to cheat the needle height a little bit trying to solve the stitch problem).
Check the needle height, meaning make sure there’s no crud up in the slot for the needle and that the needle makes it all the way up to the stop;
That fixed my problem on straight stitches, but not running ZZ. Based on someone telling me once that on ZZ the timing has to be backed off a little I started looking for how to adjust the timing. I wasn’t really seeing anything obvious so I started loosening screws, beginning with the one circled in the second picture. This did not allow me to adjust timing but what it did do was allow the shaft and bobbin case housing ( single arrow) to shift back and forth (double pointed arrow), and to rotate, as noted in the third picture. This did not allow the hook itself to be adjusted.
Having this screw loose (which I may have in general) most importantly allowed shifting of the entire bobbin case housing and hook distance relative to the needle.
Once I figured this out I had to adjust two things:
1. Position of the bobbin case housing relative to the feed dogs (top marks in picture 3). I didn’t notice the positioning of this when I first loosened the screw in question, so I had to look at another machine to get an idea of where the housing should be positioned relative to the openings in the feed dogs for the needle to come through.
2. Hook distance from needle; Picture 4 shows the hook/needle (on a non ZZ machine) distance, the hook has to come very, very close to the needle, as in the picture. This took a little bit of fiddling because I was adjusting this distance as well the position described in #1 at the same time and then trying to get the screw tightened when I thought I had it. Once I had the hook where it needed to be the machine sewed well, both straight and ZZ (although I had to back off the upper tension to get the ZZ stitches looking nice, which hasn’t been an issue on other ZZ machines).
I believe with the machine in my picture someone loosened this screw and the hook was too far away from the needle, or it came from the factory that way, such that it couldn’t properly pick up the thread loop on the upstroke, and was thus skipping stitches both straight (when initially tested) and ZZ.
Take away for you: check the hook distance from the needle, both in straight stitch and ZZ, and if it’s too much consider adjusting it.