To train a mechanized infantry, the strongest infantry unit in the game, you only need 24/68 (35%) of the techs in the tree. Here's an incomplete list of techs that you DON'T need to train a mechanized infantry:
Mining, Wheel, Bronze Working, Iron Working, Construction, Engineering, Machinery, Gunpowder, Metal Casting, Ballistics, Rifling, Steel, Replaceable Parts, Combustion.
Just for fun I tried to think of a way to make mechanized Infantry without any of these technologies. For the sake of argument I'm going to use the general sense of mechanized infantry being "Mechanized infantry are infantry equipped with armored personnel carriers (APCs) or infantry fighting vehicles (IFVs) for transport and combat" rather than mechanized infantry as we typically understand them in our current timeline.
For starters without metal casting or any hard metals we're going to have to completely throw the idea of an all metal vehicle. Metal may only be used in small imperfect quantities from what one would be able to recover on the surface (since we lack mining). That leaves us with wood, rubber, basic plastics (we lack machinery and combustion to refine it), stone, fabrics, and livestock for primary building materials. Originally I considered some kind of wagon vehicle but then I remembered I had no wheels, so instead I considered a form of low flying armored war balloon, but without a combustion engine to generate hot air that idea would have to be nixed. Then I remembered something important, I AM allowed to use electricity and there ARE surface deposits of copper and gold. So I revisited the war balloon idea with an engine that ran on burning kindling/gas via the use of heat conducted by running an electric current through low grade imperfect copper and gold rather than a petroleum or coal based combustion engine. The balloon however would make for an easy target to firearms or arrows on its own (you said we don't have gunpowder, not necessarily that they don't, so we have to consider the possibility of their shooting at us) which is why we'd need some kind of armor. I considered just declaring it Kevlar but while machinery isn't TECHNICALLY required to manufacture Kevlar the amount of time it would take would be beyond reason and it would be impossible to make the large sheets required. So then I considered alternating layers of highly tensile/flame resistant wood and similarly durable cloth, but by doing that the weight would be increased which would necessitate a larger balloon which would need more armor and you'd have this infinitely escalating arms race against its own design. It was at this time that I realized that the balloon doesn't need to fly, it just needs to reduce the weight of the craft enough to make it mobile, it's the basic principle behind hovercraft which are APCs in their own right which is what I was trying to design in the first place. So by mounting the balloons within the main carriage of the vehicle or in side mounted pods and making the main body out of highly polished and durable wood you would in essence have an ancient era hovercraft. Then came the problem of propulsion. I thought about livestock pulling the hovercraft around but it didn't feel in the spirit of the thing since it's no longer mechanized. I also thought about sails being used to guide it but then you'd reach the same problem of making a large protruding target right where the balloon would have been anyway. Then my mind fell on what real hovercraft use, propellers, but I had to come up with a way to use propellers without a combustion/metal engine. I of course could just have a bunch of dudes cranking its axle to make it spin but that wouldn't reach the desired velocity, I'd need something faster. Well, since I was inflating the balloons with hot air created by gas being burned by electricity I realized I could just funnel the runoff hot air from that non combustion non mechanical engine to the axle to spin it thus creating propulsion and, with the use of a rudder, steering.
And there we have it, mechanized infantry which don't use "Mining, Wheel, Bronze Working, Iron Working, Construction, Engineering, Machinery, Gunpowder, Metal Casting, Ballistics, Rifling, Steel, Replaceable Parts, Combustion" You see, with just a touch of imagination, a complete disregard for practicality, a ton of mental gymnastics, and a very tenuous grip on what words mean, anything is possible! I'm sure smarter minds than I could come up with even more practical or functional designs, that's just what I came up with after an hour of mulling this over in my head.