Let's address this one point at a time. First, the pinion bearing is the part that failed. Is it (really, I mean was it) a double-row bearing? That would be a cheap way to avoid the whole preload issue. It doesn't tell us why it broke up, but for the moment we can hope it was just lack of lubrication at the Honda factory. You need a new bearing of course - please post a picture of the numbers on the side of the outer ring of the old bearing if you can.
Second, there is the issue of how the clearance between the crownwheel (the large bevel gear) and the pinion (the small bevel gear) is managed. The pinion is in a fixed position. We haven't yet seen how its location is determined, it may use something rudimentary like a circlip at one end and a shoulder in the housing at the other end. Please post pictures when you dismantle the pinion and pinion housing. The crownwheel is pushed in the out-of-mesh direction by a spring, and its actual position is set by the flanged aluminium block behind the crownwheel's support spacer. Note the grubscrew in that spacer which keeps the axle shaft from slipping through and getting out of position. It appears the gear to gear clearance is permanently set by the flanged block and the width of that spacer. There may have been selective assembly at the factory, probably by choosing the correct spacer to get the required clearance, but it is not really critical, given that the parts rotate rather slowly and are not going to cover a whole lot of miles in the life of the mower. It seems a simple and effective design which should be highly reliable. Nothing needs to be done to it except clean it all up properly to remove the swarf that will currently be in everything. When you put it back together, the mesh will be right, as long as you don't make any mistakes.
I do not think the pinion bearing failure was caused by incorrect clearance between the gears, I think it began with failure of the pinion bearing. I think if there had been lack of clearance between the gears it would have been likely to result in heavy wear on the aluminium block where the crownwheel support spacer bears against it, and that wear would quickly have provided the necessary clearance. Note that even if everything is working well there must be some wear on that block, because the spring pushes the crownwheel against it. Furthermore in its last days the breakup of the pinion bearing provided a large supply of swarf which will have embedded itself in the crownwheel's teeth, and that will have caused heavy pressure trying to move the crownwheel away from the pinion. Hence there will be signs of excessive pressure between the spacer and the aluminium block anyway, due to those last days when the gears were swarf-lubricated.
My guess is that what you have is a transmission design that was probably very reliable indeed, but had the disadvantage of only providing one speed. American mower buyers seem to be greatly attracted to a choice of speeds, so Honda provided 2 speed, 3 speed and ultimately hydrostatic gearboxes to give the customer what he or she wants.
