OK. I'll play:
1. .44 Remington Magnum ammo weighs more and takes up greater space on your belt or in your pockets...so you theoretically would be unable to carry as many rounds for the long term. In reality, most folks carry a couple of reloads and call it a day.
2. .44 Remington Magnum / .44 Special ammo generally costs more and is not as widely available as .357 / .38 Special. Ammo replenishment might be more difficult with the larger caliber. In the short term, when the public goes on a panic buying spree, .44 is usually left on the shelf along with other less common calibers. The most common service or CCW calibers (.380, .38, .357, 9mm, .40, & .45 ACP) seem to vanish from shelves first. Long term...you might have a greater problem replenishing .44 after several months of civil unrest where folks are actually expending bullets in quantity.
3. .44 Remington Magnum ammo is not nearly as controllable (fast followup shots) as .357. (
On the other hand, .44 Special is...) This ability to manage followup shots under pressure is hugely important. It's what makes a high capacity 9mm (with modern expanding ammo) such a deadly package. Lethality, Capacity, Accuracy, Low Recoil. You give up two of those with a .44 Magnum.
4. A handgun chambered for .44 Mag is most likely gonna weigh a
bit more than a comparable .357... unless you go with an alloy revolver...in which case see point #3 (in spades). More weight to carry on your belt might be an important consideration.
5. Ask yourself why .44 Magnum never caught on as a serious combat caliber for military or law enforcement agencies back in the revolver era? Not to say that the occasional real-life LEO didn't pack one, but seriously...Harry Calahan was a fictional character. Full house .44 loads deliver more Sturm und Drang than is actually required for an anti-personnel scenario. If you load .44 down to controllable levels (or simply use .44 Special), you are back into the old heavy/slow vs. light/fast argument. The .357 has an arguably decisive street track record not entirely replicated by .44 Mag. Not to say that .44 won't work (see point #6 below).
6. There isn't going to be a significant terminal performance difference between the two calibers (both will do the job), but you are going to be more likely to miss with the .44 after taking the first shot.
7. With heavier weight .44 bullets, over-penetration could be an issue (if that's important to you).
8. Both calibers can be defeated by soft armor, so .44 doesn't offer any advantage there.
9. If you envision your life and death scenario happening while driving, certain .44 loads offer good penetration on auto bodies and glass.
10. Muzzle flash is a consideration for both loads, but especially with factory full house .44 hunting ammo. Night vision is nice to have and keep in a fight.
11. A 629 loaded with a good .44 Special hollow point is a very viable self defense combination. It fits into the time proven large bore defensive revolver club (.45 ACP. .45 Colt, .44 Special). Big bullets moving at low or moderate speeds have worked well for over a century and a half. Stoked with .44 Specials, that big 629's recoil would be light. Good accuracy, lethality, and control.
Lest you think I'm going all "negative vibe" about your possible .44 choice, I'll mention that I occasionally carry a 4" Model 58 .41 Magnum (loaded with mild 175 grain Winchester Silvertips). I just happen to like the caliber and the gun. If that 629 really makes you happy, then get it. With adequate range/hunting practice, you'd certainly be well armed.
But honestly...A good .357 duty revolver is just a better choice
for fighting than a .44 Magnum. .44 is a better caliber for hunting.
Your's is a nuanced question with no truly right or wrong answer. If you can hit accurately and repeatedly with the .44 while you are under stress...then it's the right choice for your purposes. The .357 is technically a better choice. But a man armed with an old Colt SAA would also be well armed as long as he really knew how to use
that weapon.
If you just want a .44 Magnum for self defense...get one. You don't need logical reasons for choosing that caliber for combat use. Which is fortunate... because there really aren't any...unless you have a need to put down zombie bears.