There are no attributes for a player. There needs to be a dominant offense, a dominant defense, and an offensive coordinator that likes to run near the goal line. And the coach can't have too much trust in their QB because RBs scoring receiving TDs is more "luck" than skill. And then add on a bit of luck on a player lasting close to 16 games... maybe on the health concerns you can discriminate against the smallest and biggest 10-15% as they are more likely to get hurt.
So four of the guys heavily benefited from Norv Turner/Marty Schottenheimer/Dick Vermeil/Sauners or their direct disciples LT, Holmes, Faulk and Emmitt. These guys would run the ball every play if the defense didn't blitz. They for sure hated throwing the ball near goal line. And to make it the perfect situation(s) in Dallas and San Diego, both teams had good QBs but not great and definitely not trusted around the goal line as a stud. If you were playing fantasy football 20 years ago, you drafted the RBs these coaches were coaching early.
Shaun's was about the same. Holmgren has always been a run in the redzone guy. Him and Favre got in fights because Favre loved stat padding on 1 yard TD throws (stat check but Favre has to be on the all-time list for 3 yard and less TD throws). So Holmgren plus Hasselbeck means big numbers for Shaun. Let's not forget that during Shaun's three big years the Seahawks had THE BEST offensive line in football with two future HOFers (at least two).
Faulk's big season is easier. It was a top 5 offensive line and maybe the best offense ever in the NFL (adjusted for year). Faulk just had a ton of chances and didn't get hurt. Martz/Vermeil did throw it a ton around the goal line, so it made the runs even easier for Faulk.
Going back to LT he was on a run first team that had a great offensive line and two pro-bowl skill players that were known for blocking - O'Neal/Gates. But LT was arguably the best goal line runner ever. He set records at TCU before coming to the NFL.
So to break the record you need a lot of help from coaches and teammates and probably need to be one of the best ever at RB. LT in the list you had was the best all around back, Faulk may have been a better receiver but wasn't close as a runner... and maybe Emmitt is equal as a runner but no way at receiver. Also didn't LT throw 2 TDs that year (I had him on my fantasy teams - draft Marty backs- so yea I know he did).
Note: This record could be broke again. You are exactly missing it with the reasoning that it can't. It might be harder now. The issue is that the NFL has made passing easier. This means more passing TDs so less running ones. But the biggest issue is the offensive line. If you are a road grating OL you aren't as valuable in today's NFL. Passing is easier so GMs are more likely to draft a two-point stance 6-8" guy with super long arms than a 6-4" guy that will bowl people over. The 49ers are a good example this year. They run the ball really well and if they had a HOF back maybe he gets 20-25 TDs total... they purposely went out and got all of the smaller leftovers that can run block that other GMs didn't want.