Anywho, getting back the to OP's question...
I have owned and shot both 44 special (Ruger flat top Bisley, 5.5"... sold it)) and a 44 mag (S&W 629-3, 4". Still have it). Never took a deer with either of them.
I reload to modest 44 mag velocities. NOE 44-265 cast to about 1000-1100 fps in both chamberings. Easy to shoot, accurate, and a presumption they would take a deer at reasonable hunting distances for me in my Maine woods ( 25 - 75+ yards).
Staying sub-sonic in MV extends the distance for effectiveness, as you don't have to have the bullet pass into, or back from, super sonic flight (avoiding the trans-sonic zone). It's good for accuracy too.
I really liked my blued Ruger Bisley Flat top 44 special, but it was an extra gun that didn't do anything that my S&W 629-3 (with 4" barrel) revolver couldn't do, so I sold it.
I'd rather have a modest 44 mag load (~ 1100 fps with 240-260 grain bullet), or a similarly loaded 44 Special, than a heavy, hot-loaded 357 magnum. I agree with the OP. Nothing wrong with your heavy 44 special load, and a lot of things very smart with it. Go hunt with confidence with your set up.
Some good 44 Special reading: Enjoy
Handloader magazine 44 Special Pet Loads. (Brian Pearce)
http://www.goodrichfamilyassoc.org/44_Special_Articles/Brian%20Pearce%20on%20the%2044%20Special.pdfHeavy bullets in 44 Special ( Ross Seyfried)
https://www.docdroid.net/ulji/ross-...p-44-special-bisley-revolvers-pdf#page=2