Large milk dough for pizza; doesn't seem to knead at all

Probably a combination of causes. You're using whole wheat flour, which really does not knead like white flour, even used in moderation with other flour: in addition to having less gluten, the shards of bran cut through the dough structure as you knead. Particularly if the "refined flour" (hmm) you were using wasn't high-gluten bread flour, that could leave you with very little cohesion. The olive oil will also interfere with gluten formation, as will the milk to some degree, particularly whole milk.

Now, none of that may actually matter. Pizza doesn't need a gluten matrix for structure in the same way a free-form bread loaf does, and the recipe you've chosen indicates that you're not looking for the classic Neapolitan crust texture anyway. If you're forming the pizza on a pan, you don't need the gluten to pull the crust, and the starch gel that forms during baking will make it hold together just fine as a cooked pizza. Treat it like pie crust, and roll it out instead of pulling it out.

Oh, incidentally, 10 tsp is a real weird measurement. It's just shy of 1/4 cup; use that instead, and give your measuring hand a rest.

I would say that you simply didn't knead enough, and modeled your expectations on siding that are not similar enough.

As Sneftel noted, the whole wheat flour is part of the difference. I'm not sure it has less gluten overall, but it is more difficult to develop, first because of the cutting effect mentioned and second because the bran soaks up a lot of liquid, leaving less available for gluten development. 66% is lowish hydration for a bread with significant amounts of whole wheat.

The main problem is, in my opinion, insufficient mechanical action. Kneading dough is hard work, literally. Modern home bakers accustomed to small batches of rich white doughs might not be as aware of it, but actually dough needs quite some pounding. The batch size absolutely matters - if you sink the same amount of mechanical energy as usual into a batch of doubled weight, you will need to knead twice as long to get the same results. Add to your doubled batch the fact that you are working with a dry whole wheat dough - not only is there less chance for gluten formation due to the missing water, but the firmer dough means your physical punching and pulling doesn't transmit effectively through the dough.

In general, if you are working a mixture of water and wheat and it is not yet forming gluten, the answer is pretty much always that you haven't worked it enough, no matter what the clock says. Dough has to be kneaded until it is properly developed, and you only need to give up and suspect foul play if it turns into overkneaded state without noticeably passing "proper dough territory" on the way.

After a few days of retrospection making the pizza from this flour and also making a fresh dough, I think I have an answer to my question. There are a few points that come to mind:

1. As I said in the question, the dough wasn't sticky. I repeated this recipe minus the whole wheat and milk(this time I used 1.25 cups of only refined flour with ~120 ml of water and no milk) and even this time it wasn't sticky. Upping the water content to 150 ml helped the issue. Adding more water made the dough easier to knead and also caused it to break less when stretched.

2. When I actually made pizza from the bread, I found that it had bubbles in the dough, but it was more of a cakey than bready texture. I guess this was because of the milk. Adding milk is something I'd avoid going forward, even though it helped the crust brown a bit. It made the dough sweet(a contradiction to the sour flavor of the sauce and cheese) and it made the dough more like a cake than a bread.

3. One point that I realized when making a fresh dough ball is atmospheric temperature and humidity. Temperatures during daytime range between 25 and 30C and humidity is low, which might've contributed in making the dough even drier than it already was. Might need to compensate for that in summers as compared to winters by adding a bit more water.

To summarize, this is what I'd do next time:

1. Only tweak one thing in the recipe at once. I tweaked multiple things(multiply the recipe by 2.5, add half milk instead of water, use part whole wheat flour instead of all refined flour).
2. Add water according to the recipe, but if isn't sticky(above 60% hydration ratio), then add water till it is.
3. Adjust recipe according to atmospheric temperature and humidity.