D&D 5E How many attacks should a two weapon fighter get?

[S'mon]

But as I understand it, rapier + dagger was developed for use in situations where carrying a light weapon or two would be acceptable, but not a heavy sword and a shield. I would expect that between two equally well-trained warriors, one focusing on rapier + dagger and the other on broadsword + shield, the dual-wielder would be mincemeat.

Yes, IRL of course people did not dual-wield on the battlefield; it happened on Venetian city streets
because sword & dagger was comparable to sword & buckler and either could be carried going about daily business. Neither compares to sword & large shield or 2-handed polearm.

For gaming, I like my 2WF Rogue types, Fafhrd & Mouser, to be viable (and I think 5e does that fine - 2WF Rogues get a better chance to get a hit and thus inflict their sneak attack damage) - but the heavy armour 2WF Fighter is a uniquely D&D trope I think, an artifact of 1e's over-generous 2WF rules (try playing a twin handaxe
weapon specialist 2WF in 1e-with-UA, it's totally insane!). I like my serious Fighters sword & board or great weapon, if they 2WF it's because they're showing off, not for practicality.