Following the usual rule of thumb, the 6mm thick wood suggests 2-3mm dowels are as large as I should go (this is about 1/16 inch).
3mm is maybe pushing it (as it leaves only a quarter of the wood's thickness to either side of the drilled hole) but on the other hand 2mm dowelling would be really skinny, towards the low end of what's really practical in a lot of woods — could be excessively prone to breakage both during manufacture and usage.
If you're comfortable using 3mm I wanted to mention that you can use bamboo skewers. These are especially strong, and from the sets I've bought typical diameter seems to be in the range of 2.8-3.1mm. Because of the sometimes sloppy fit in a 3mm drilled hole I would advise switching adhesives and using epoxy because PVA has almost no gap-filling ability.
1/16" is 1.58mm BTW. 5/64" is a closer approximation to 2mm (only 0.02mm off), and although this probably seems an awkward fraction to a metric thinker it actually appears to be a commonly available bit size.
But wood glue is pretty thick and gloopy (viscous), so how would I be sure to get good glue coverage.
You'd use a suitable glue applicator, same as when applying glue to any recess where you want fairly uniform glue coverage.
I was planning on coating the dowel and pushing it in, hoping that would give coverage down the full length of the dowel.
Honestly, for these that's likely to be perfectly adequate. After all, some woodworkers only apply glue to one side of almost every joint!
It would be best practice to add glue to both the hole and to the thing going in the hole (just as it is in mortise-and-tenon joinery) but I have seen other people barely glue theirs in. And it's worth mentioning that in some applications wooden pins/pegs (typically sharpened to some kind of point) are used like nails and aren't glued at all.
However I like to be as conscientious as possible and glue mine in, because, why not?
BTW if you want to maximise the reinforcement pins provide, don't put them all in perpendicular to the surface. A slight angle on some or all — e.g. like this / \ or this \ | / gives a dovetail-like hold, which gives a noticeable bump in strength. As I mention in a previous Answer or two, the angle only needs to be shallow for the effect to kick in, so the ends aren't very obviously oval in case that would bother you.