Semi-seriously... nope, can't be serious. -20F and riding bikes? Nope. Nopittynopenope. Also nope.

Only use I can imagine for a handlebar mounted doodad in winter might be for navigation and emergency comms, for bikepacking or long rides in unpopulated or lightly populated areas. There's nothing like that under $50. A phone could substitute and could be used in a handlebar bag protected in a map pocket with a USB battery in the bag for extended runtime. And Topeak and others have weatherproof mounts/pouches for phones. It might be possible to squeeze a dry chemical heat pack or similar doodad in those to keep the phone warm enough to run reliably in subzero temps.

Also I'd check to be sure there are lithium batteries available for any device. Those may perform reliably below zero. I used lithium AA in winter for my cameras and flashes that took AA and they worked fine into the teens, which is as cold as it gets in my area (traveling between Texas and Oklahoma panhandle). NiMH batteries were okay down to the 20s. My cameras that use rechargeable proprietary Li-Ion batteries tend to work at all temps I encounter. I've even stored 'em in the freezer overnight to test, but that doesn't get anywhere close to zero, let alone -20F.

Any wireless speedo/odo will use at least two batteries, often different types -- one in the unit on the fork, another on the display/receiver. If they both use lithiums they may work in subridiculous temps.

There are a zillion speedo/odo units on Amazon and elsewhere for under $50. I got a "SY" branded basic wireless computer for $12 in July -- the same unit is rebadged under various names. Works fine, weighs almost nothing, does what I need, and I won't be upset if it breaks or gets stolen. It's always virtually identical to my phone apps.

But I use the little speedo/odo only on my road bike for serious workouts to push myself. Can't say that it's actually helped. All my PRs and my one or two KOMs were sans-handlebar mounted doodads, back in the spring. I stripped off everything unnecessary to save weight and aero drag. Just the phone in my jersey pocket or saddle bag.

On my hybrids I only use my phone, which is in my pocket. I don't really need to know I'm loafing along at 12 mph.