The main thing was cost and ease of manufacture. For this post I will ignore all except the German and US lines, other nations followed one or the other's line of development.
Germany: Designed from scratch. Front (steering wheels) unpowered, thus no need for a steering axle transmission. Steering was done like a truck up to a point after which further turning of the steering wheel applied a brake to the track on the side to which you wished to turn. The track suspension while using interleaving wheels, unusual for the late '30s, really is nothing particularly complex, just more wheels. The engine drove a transmission that only drove the tracks, with the only complication being the hard steering braking. The tracks were steel plates with rubber pads, connected by pins seated in oil filled needle bearings which produced less 'drag' than plain shoe and pin types but had higher maintenance. The front axle being unpowered was more akin to a simple truck front axle and cheap. Most nations' truck/bus manufacturers were used to unpowered front wheels. Germany was no different and seemed to unable to convert all lines to all wheel drive (4x4 or 6x6).
US: The US basically took a 4x4 chassis (M3 Scout Car) and replaced the rear wheels with a Timken (a development from the Kégresse system) bogie unit and fiddled about with the hull/armour arrangement. The front axle was already powered, so better than the German equivalents for ditches and low obstacles, but with the shorter track a bit more likely to bog. The track mounting and bogies were somewhat flimsy and the track which was a one piece rubber moulding with an embedded steel ropes and plates "ladder" wore and shredded far easier than the German ones. The US with it's industry was far better able to produce all powered wheel vehicles (4x4 or 6x6) with less hassle. Unlike the German H/T, track steering didn't contribute to vehicular steering.
The Germans could be said to have taken a simplified tank track transmission with a long track base and added a front unpowered truck steering system while the US took a 4x4 and replaced the rear wheels with a shorter track unit.
As for steering wheel vs levers: Levers are simpler to implement for tracked vehicles, though the Tiger I was one of, if not, the first to give the driver a (truncated) steering wheel. As for the US and German H/T, the initial (German) or all (US) steering was done using a truck type steering axle (US: powered, German: unpowered), so a normal steering wheel was the sensible choice from both the production and training aspect.
As you can see, H/Ts used as many old components or designs available to produce the cheapest vehicle to do the job. By war's end both nations had intended to replace H/T with fully tracked vehicles due to their inability to consistently keep up with tracked vehicles over all ground conditions and obstacles. Better than trucks but not as good as tanks, mobility wise.