I've got a 77/22 in 22lr, and have fired the 77/22Hornet and the 77/44. They're sweet handling little carbines.
A .357 carbine would suit my purposes better than the .44 and yes, it's plenty good enough for deer too if you use common sense with range & shot placement.
Load it up with Sierra 170gr JHPs or your favorite 180gr cast lead bullet and take it after deer.
I do like the little Marlin lever guns in the revolver calibers though...those are super little carbines too.
As for being able to work the bolt fast enough for follow up shots...it can be done easily with most bolts. My favorite for fast bolt work are the cock-on-closing Mausers but a good SMLE is just as slick. Cock on open actions might slow ya down a bit compared to cock on close but enough to prevent a quick follow up shot.
I've tried running the bolt with my off-hand but it just isn't as natural for me as the guy in the vid makes it look. It works with short actions, but with a long-action you will punch yourself in the snout.
I still prefer the old reach across trick. With iron sights, there's no scope to get in the way and the bolt handle is right there without dropping the rifle off my shoulder. With a scoped rifle, I do cant the rifle away from me when I'm reaching for the bolt handle but the butt never leaves my shoulder. I grew up a lefty that learned to shoot with righty actions. Have only had one left-hand action in my entire life and it wasn't worth the extra cash I spent on it.