You can get 38 Specials that function in the `73 just fine, but the need to be of this variety/OAL:
http://www.midwayusa.com/product/16...58-grain-lead-round-nose?cm_vc=ProductFinding
Regardless of the OAL, you might want to change your link since you accidentally linked a cartridge with a round nose bullet. This is generally a well recognized safety no-no for any tube fed lever gun.
The issue is of course that the nose of each round is pressing against the base containing the primer of the round in front of it in the tube. According to how the action works, the rounds can actually jar quite a bit against each other as they move back in the magazine.
The toggle link action of the 1873, 1866, and Henry with their straight vertical cartridge elevator may be the worst for this. Enough so, that set back of the bullet into the case can occur if the crimp is not good. That's bad enough with out having a round or multiple rounds possibly detonate in the magazine.
Primers are, or should be, slightly inset into the primer cavity, so a flat nose helps insure that no contact can be made. Even when I have bought factory ammo, I always double check the primer seating---especially if I am going to use the rounds in a lever gun.
Magtech makes several FNJ flat nose rounds plus flat nose solid lead bullets in their cowboy action line.
As far as OAL, you will have to see what works and what doesn't. I've had no problems as long as I can hit at least a minimum of 1.43 when reloading a .158 grain flat nose in my 73's. I recently picked up a older Rossi (pre-bolt safety) in .357 mag and although I haven't shot it yet, those same reloads seemed to cycle through it just fine.
Most factory ammo I have bought was at least 1.44, so I've had no issue with that.
Cheers