Several points here:

Axle diameters and threads vary; yes some hollow cones will fit some solid axles, but the curves of the races determine the angle between contact points, which must remain between narrow limits for the bearing to have a good service life. For this change get the best match you can.

You don't get a lot of extra strength from a 3/8" solid axle compared to a 10mm hollow one - about 15%. I suspect that unless you get a high-quality solid one, any good quality hollow one will be of stronger steel and better finished sufficient to outweigh that 15%. It's the outermost portions of the section that contribute most; hence Increasing the diameter makes a relatively bigger difference - the Atom front hub that uses a solid rear axle has almost double the strength of the normal front axle, this just going from .3125 to .375. Common solid rear axles are smaller than QR ones (they are .3937...); you lose a little of this disadvantage as a 26 TI thread is a little shallower than a 1mm thread. If possible use an axle without a slot.

Axles fail by buckling. The compressive force necessary to fix the locknuts against the dropout faces is not enough to cause buckling by itself, but it will reduce the size of the side-load imposed by the bearings required to initiate buckling. QR's should be tightened enough to keep the wheel from slipping, and no more.