Quoted:
The lead on .22s is too soft to push through a loading gate.
No, it's not that simple. The reason that no rifle manufacturer makes a 22lr rifle with a side loading gate is because of the basic weakness to side loading the 22lr with it's heeled bullet design has. Unlike most other cartridges where the bullet is a smaller diameter than the brass case it fits into, the 22 family (22lr, 22long, 22short, 22 CB, and 22 BB. Early cartridges were designed with heeled bullets so that old, black powder cylinders could be used by simply cutting off the rear, machining in a recess for the case rim, and then adding a different hammer.
The issue with a heeled bullet is that very little of the bullet need be inserted into the case for the cartridge to function. This is the case with the 22 family of cartridges, however, this leads to a basic weakness to side loading. It is easy to push a 22lr bullet out of it's case simply by pushing on the side of the tip. When removed, one can see how little of the bullet was inserted into the case. So, the act of pushing the round through a loading gate would result in damaged or separated bullets thereby causing failure to feeds, jams and misfires.