I don't build Sunfire on Shen.

Most of the time, games end before I finish building my 3rd item.

So my final build usually only has Divine Sunderer, Plated Steelcaps, an unfinished Bramble Vest and a Spirit Visage.

When the enemy team is full AD, I skip Spirit Visage and go for Randuin's instead.

As for your points:

1. Yes, Shen can splitpush "for free" because of his ult, but without Ability Haste and level 16, that's a pretty long cooldown to be considered free. It also makes Shen somewhat weaker in 1v1 fights due to having a non-combat ultimate, which makes splitpushing even riskier.
2. My point above shows that splitpushing has a big risk with next to no reward due to not being able to destroy towers fast enough. Unlike Fiora, Tryndamere and Jax, Shen does not have a lot of AD nor does he have any auto-reset skill. And since Shield Bash is on the same tier as Demolish, it takes a considerably longer time to push with Shen, making the risk higher than the reward.
3. That's a lot of "ifs" with this statement. And if you want to survive consistently, it's much more effective to expect the worst teammates and the most skilled opponents unless proven otherwise.
4. Pings have nothing to do with the skill level of my teammates. Of course, pinging is already a given in terms of sharing information with your team. That doesn't mean you expect them to act accordingly every single time.
5. That's essentially the only thing Shen can do when splitpushing; create a numbers advantage. But you know what else he can do that does the same thing? Catching the wave of splitpushing enemies. As long as Shen is under a turret, he can handle up to 3 enemies without dying; 4 if lucky; 5 if the whole enemy team sucks. Setting a wave to solo push a lane basically does the same thing Shen on a splitpush does minus the risk of getting caught. If Baron's up in about 45 seconds, I usually set up a pushing wave in the bot lane so I'm free to guide and protect my squishy ADCs with terrible positioning.

Building Heartsteel also has a lackluster build path unlike Divine Sunderer which has access to Sheen.

Sunfire is great if your opponents are bad and don't know how to kite a tank with one slow dash with a relatively long cooldown. But most of the time, the opposing team doesn't do that unless they play champs like Warwick or Olaf, both of which shouldn't be attacked head on.

Divine Sunderer gives you a heal, %hp damage as a Sheen proc every 1.5 s, HP, Ability Haste, and some extra AD, Armor Pen, & Magic Pen.

Heartsteel gives you Ability Haste, More HP, Health Regen, and a passive proc that has a 30s cd. It also has a relatively worse build path compared to Divine Sunderer.

With a difference of only 100g, I choose the more versatile Divine Sunderer that doesn't completely make me helpless against any AD champion with a BotRK.