Whether top rope soloing or lead rope soloing, crossload prevention is an important consideration. But surprisingly, it is for two quite different reasons:
TRS: Better Feeding
For top-rope soloing, the forces on the progress-capturing device are relatively low. In this application, a crossloaded carabiner is unlikely to fail simply because it is crossloaded.
But crossloading still isn't ideal. A device that can rotate, shift, or sit awkwardly against the carabiner can add a bend in the hanging rope. This disrupts auto-feeding. Keeping the device oriented correctly can make the system move up the rope noticeably smoother while being easier to manage mid-climb. (Keep the fixed rope between your legs, too, for the straightest rope path!)
So for TRS, the main benefit is better feeding up the fixed rope.

LRS: The Critical Connection
Lead rope soloing is different.
The connection between your progress-capturing device and harness can experience much higher forces, along with loading scenarios that are harder to predict. And you often have one critical carabiner connecting to the belay device.
That’s the connection we want to keep oriented correctly. It is unnerving to see the belay device sitting sideways, rotating onto the gate, or ending up in an unintended loading configuration. It will probably pop into a clean axial position in the carabiner if it is loaded during a fall, but that sideways device is mentally distracting while trying to focus on difficult free climbing.
Lead rope soloing is already a fringe use of climbing equipment. Even a well-designed system isn’t going to behave as smoothly as a partnered belay, and there are failure modes and awkward situations that simply don’t exist with a human belayer. Adding crossload prevention is an added layer of control and peace-of-mind to the system.

Harness Side vs Device Side Protection
Many carabiners come with built-in anti-crossload features (see the small silver wiregate in the blue photo above). It is usually on the small zone of the pear-shaped carabiner, which still gives the connected device a lot of movement across the larger curved side of the biner.
As mentioned, as soon as a load is applied, the device should pop into a nice axial orientation. But we always find a sideways-sitting unweighted device to be distracting while free climbing.
So for lead rope soloing, we like to double up anti-crossload protection to have both harness-side and device-side protection.
(Yes, we have tried the DMM Rhino, it almost provides built-in device-side protection, but the unweighted device can still sit sideways near the nose)
Carabiner Size Considerations for Device-Side Prevention
Using device-side crossload protection with a carabiner that’s too narrow can make it difficult to open the gate and install your devices.

We prefer wide oval and pear-shaped locking carabiners. They give the progress capturing device and Flex-Link more room to sit naturally and make installation much easier.
Additionally, I-Beam style carabiners give the benefit of effectively snapping the Flex-Link "into plane" as opposed to rounded carabiners that are slightly less secure. The Flex-Link stays flat inside the biner and won't snag on anything.

We've really enjoyed this BD Vaporlock Twistlock Carabiner for Top Rope Solo Setups
These small details can make a noticeable difference in your rope solo experience. You'll worry less about equipment and focus more on the climbing movement.