That recipe actually says there are two ways you can prepare this:
There are two ways to do this: throw just about everything in a pot of
water and simmer, then add the milk, tea and sweetener and brew for a
few minutes; or, let the spices soak overnight in the water then
simmer in the morning with your tea leaves and milk.
(emphasis mine)
Have you tried the latter? That way you still only have to heat things up once.
I'd be hesitant to boil milk with spices and then store it cold, reheating again. If you're using dairy milk, that is. If you're using anything other than non-fat (skim) milk, you'll get a "skin" on the top of the milk when you're boiling it - that's one of the classic signs of masala chai for me. But anyway, it becomes substantially less appetizing if it has time to cool into globules and is then reheated again. I'm not sure what problems, if any, you'd encounter using non-dairy milk like almond or soy.
You could dump the pre-ground spices into your brewed tea with milk and sugar, but the flavor profile will be a little different. Boiling the spices with the milk and tea gives everything time to meld nicely and provide a really smooth flavor blend. But I don't see any especially compelling reason that you couldn't do it.
I would like to point out, though, that boiling enough water for a travel mug's worth of tea doesn't take any more time to do on the stove than it does in my electric kettle. Adding milk shouldn't make too much difference; I'm willing to bet that after you get familiar with the routine, you can follow this recipe and still have it take 5-6 minutes. :)
Purists will insist that the spices must be boiled with the water, but those are the same people who would scoff at you adding cold milk to your tea in the first place. There's no reason why you can't, other than it might cool down the rest of your tea.
I've never had an issue reusing anything that's real tea (black, oolong, green). Perhaps there is another factor here? If you're black tea is coming out of a tea bag, or a lower "grade" then the green tea leaves you're using that might be a reason that it isn't coming out desirably.
Generally speaking it's an issue of grade rather than the type of tea. The finer the tea, the more likely you are to extract everything in the first run. Take apart a cheap tea bag, and you'll notice that the leaves are powder or almost powder. A full leaf (you can buy any black tea from a reputable shop in a higher grade) will extract more slowly and give you a better flavour and probably yield to better second or third runs.
Best Answer
Not sure if this is the situation in your case, but here's a possible stab. I hope it's informative (and correct!) anyway. :)
Depending on the look of the leaves (can't tell from your link), you might have just the tips and buds, rather than fully formed and unfurled leaves. If this is the case, this can indicate higher quality (or at least more desirable, commercially speaking) tea, and perhaps the time of year of harvest. If very young, these buds won't unfurl much (or at all).
When harvesting tea, one could pick the more mature (older) leaves further up the branch (toward the trunk of the tree), or further down toward the end of the branch (toward the newest buds), which will be smaller and more delicate. The latter (buds and smaller leaves) are "better" in various respects, and are generally more expensive in respects of harvesting, processing, etc., and correspondingly less common and more valuable.
Take a look at tea leaf grading guides; the Wikipedia article on tea grading for more on the concept. There are many more guides if you search on that term. Though usually applied to black teas, I hope it illustrates the point I'm trying to make. Grades of "FOP" ("Flowery Orange Pekoe") and higher will have some quantity of the buds; more words means "better" in some regard (Special Finest Tippy Golden Flowery... I have heard of SFTGFOP1, but I am certainly not refined enough to appreciate it). Good white teas (e.g., the picture on this page is an example of leaves that are "hairy" because the buds are just opening) sometimes have lots of buds, which won't unfurl when steeping. Again, you're talking oolong, but hopefully this illustrates the concept.
Other teas are mechanically rolled (e.g., gunpowder green tea) so the look of the leaves before steeping could be similar between buds and processed teas. Gunpowder tea (the stuff that that I get, anyway) is relatively lower quality, made from more mature leaves, which unfurl completely into large leaves after steeping.
As an aside: For good quality tea, I find using too-hot water yields less desirable tea. YMMV.
I hope you've got a great tea!