If it were the case that you could turn out new equipment with a few TL 15 makers, then there would not be anything but TL 15 worlds in the Imperium. You can plausibly argue that it should be like that, since by the time period of the game we will probably have achieved a post-scarcity society, but that is not the premise of the game. TL 15 factories for military production on lower TL worlds are a plausible partial solution, and there is support in Traveller canon, but those will only work by shipping in lots of inputs from TL 15 worlds (otherwise we are doing away with the premise that some (most) worlds are lower tech)
Indeed during the industrial revolution the pace of change accelerated. If you want to assume it will stay like it is now, we'll be at TL 1000 by the time of the FFW. Which is totally plausible, in fact in my opinion it is more likely to be what happens if mankind actually survives that long, but it is also not the premise of the game, mainly because it wouldn't be a fun space opera.
Horse cavalry was successfully used in WWII for certain operations by less mechanized armies, but its use became more and more limited, due to its role being taken over by mechanized vehicles, its vulnerability to air power, and the fact that fewer and fewer people were learning to ride horses. It was considered effective by certain combatants, though others phased it out. Not really relevant to the point I am making because 19th century examples would serve just as well for the point I am making, but I wanted to be accurate. Read up if you want to know more.
On the last point, I agree they would have stockpiles of TL 15 equipment. However, if you accept the premise that there are worlds lower than TL 15 then you seem to be asserting these worlds must have unlimited amounts of TL 15 gear in their frontier warehouses. Otherwise, their limited stockpiles will run out at some point. Or be blown up or captured by Zhodani. Once the Zhodani have your magic TL 15 warehouses that provide unlimited equipment, you'll be in trouble. However, if you put those magic bottomless warehouses behind the lines where the Zhodani can't get them, then you're back to the logistics problem - you have to ship it to the front line. More likely, you have a certain amount of gear on hand, and when you run out, or it gets destroyed, you might try to ship in more but you will get disruptions, at which point you either surrender or rely on local production.