All things are relative, but the term "cowboy load" is also somewhat misleading.
For example "cowboy loads" in .45 Colt are typically viewed as underpowered loads, especially when used in cowboy shooting sports.
But who amongst us would consider 255 grains of .45 caliber lead at 1,000 fps as "underpowered"? Because that's pretty much what the original .45 Colt loading got out of a 7 1/2" barrel.
By comparison, plenty of people today don't consider a 230 grain .45 caliber at 850 fps to be underpowered.
Lots of game was reliably taken long before what we consider to be "modern" smokeless rifle cartridges, too.
According to the internet, the original .44-40 was 200 grains pushing at 1150 - 1200 fps from a 24 inch barrel. That is the original "cowboy load" for the .44-40. If the loads you have now are below that, then I'd call them underpowered cowboy loads intended for target practice or cowboy competitions.
Hunters of days long past probably weren't much different than today...in other words, hunters tend to hunt within the effective range of their weapon. They didn't use shotguns for 50 yard shots, and if their rifle wasn't effective beyond a given distance, they worked to get within range.
There are ALWAYS more effective calibers, velocities, bullet designs/mass, rifles for a given critter. But if the vast majority of your deer hunting is in the brush where your effective shooting range is likely to be well within 100 yards...why wouldn't a .44-40 rifle with at least the original "cowboy load" be effective?
And if it's not...the cheap solution is a couple boxes of modern .44-40 hunting ammunition...enough to sight the rifle in with a few left to hunt the season.