Here is the problem with this theory, and any other goofball theories (i.e. stacking dex). It completely grates against the game's design.
there is something intuitive about pretty much every character. If you look at them they will get an "enhanced STAT" trait at level 6 and 14 or something. For WAR that stat is VIT. For PLD that stat is VIT. For LNC/DRG it's STR. For PGL/MNK it's STR. Etc. Etc.
If WAR was intended to stack strength primarily or that STR would be the WAR's intended strat, this type of thing would only be confusing and serve no real purpose.
Likewise, if WAR were "intended" to stack STR, then we would be able to roll on STR based accessories in dungeons at the least. But we can't. We're relegated to only being able to roll need on the VIT based accessories.
Some STR can help a WAR once they hit a certain HP threshhold, (Ideally over 7k), but it's clearly not designed to be the first place the WAR turns.
You absolutely need over 6500 hp at the very least for titan. If you step into coil you need even more than that.
Even if you were to stack STR/Crit to make your Inner Beasts a more reliable heal, it still doesn't scale as fast as monster damage does, and if you try to make it do so by stacking DPS stats, you will be at a low enough hp pool that you will die in burst from an auto attack crit and special attack.