On the subject of should two-handed weapons get 1.5*str, the answer IMO is a hard NO. As simple as multiplying by 1.5* is, it is an unnecessary complication in a game that tried to strip away such things. If 2H weapons need a boost, just give them a boost (2d8 instead of 2d6, or heck just an actual +2 damage).
*plus rounding, such that Str 18 has even more of a leg up on Str 16 than it already does, re-igniting the 'point buy and array don't let me start with an 18' debates.
On the subject of magic items and feats--yes, one-handed weapon, shield, and dueling fighting style works very well in feat-less and magic-item-less* games. We can go back and forth on frequencies but I doubt anyone has any hard numbers. Suffice to say, I don't see that many feat-less games. Regardless, the answer can be different between those games and others.
*Or games where the DM places magic items specifically helpful to the party, so if everyone plays a Dex-based character, there will be lots of magic rapiers and few if any magic greataxe/greatswords.
**and in those that do, there seem to be a lot more paladins and full-casters than barbarians and fighters and such
As to how they can compete, they do just fine in many instances. A reckless-attacking half-orc barbarian* with a greataxe does very well. As does the character that finds the flaming greatsword while the dex-based duelist** is fighting with the party rogues and bards over the one magic rapier that dropped. Also worth noting is that the 14-Dex-and-high-strength medium armor character can drop their 2H weapon and draw out their bow in a way that sword and board has trouble doing (because of shield don/doff rules). Honestly, greatsword/axe vs. sword&board seems like a pretty fair tradeoff. It is when PAM halberd (or, even moreso 1H quarterstaff or spear, potentially with shield and dueling fighting style) comes into play that I find them finally losing their luster.
*possibly multiclassing into fighter3 for GWF fighting style and 19-20 crits
**Yes, there are str-based sword and board options too, but 'why bother with a greatsword if you can be a dex-based duelist character and also get the 'better' stat' is something of a theme I've seen.