I'd point out though that the bonus attack would apply to all weapons, single, two handed and ranged. Only the two weapon fighter gets hosed. Not sure how to fix that.
If buffing the Champion's damage output is your intent, bonus action attacks aren't the best way to accomplish it. Something like a 4e-style slayer ability (1/short rest add 3d8 damage to an attack you make) would work, though it'd be counter to the design intent of the Champion - not having to manage resources.
As you say, it all depends on how far they lag. If the only way they can keep up is with two handed weapons, that's not groovy though either.
It doesn't exactly take a boatload of strategy to imagine that the more attacks you make, the more crits you can make. Anything that lets a champion attack more - bonus action attacks, standing in the front line and making OA's - means more crits. The archetype is flexible enough to let you trade more crits for more AC (have a shield) or more back-row ability (use a ranged weapon). If damage is your concern, whip out the light weapons. If AC is your concern, use a shield. If getting your face smashed is a concern, use a bow.
I do think they lag much farther behind than you do. Give a BM, a ranger with Collosus Slayer, a paladin and a champion the same combat stats and same weapons, the champion will always be dead last.
If that's true, then it's a trap option. Since the only things champions can do is deal damage, I would hope they're ahead of the pack, not at the back, considering all the other classes have other options in addition to dealing more damage.
This is mostly
numberwang, though. The intent of the champion isn't to have numberwang, it's to be simple. If it is simple, and it pumps out enough damage, it's successful at its intent. Again, it's like ragging on the Bard because it's not as good as a Life cleric (or ragging on the Paladin because it's not as good as the Bard). I don't need to have the highest numbers to be good at what I do.
Maybe it doesn't provide that threshold of damage, I dunno, I haven't crunched the numbers. But saying "My damage isn't as big as a hunter ranger and the smiting paladin!" is not the same as saying "My damage isn't big enough." The first problem is just one of square-peg-round-hole. I don't make an assassin rogue and get sad because the thief is more mobile than I am; I don't make a tempest cleric and get annoyed because the valor bard is better at charming; I don't make a wild mage and sigh enviously at an abjurer's AC. It's entirely possible the Champion doesn't deal the most damage in the game, and that this is entirely what was intended, because the Champion is meant to be "simple and effective" - to do its job without fiddly bits. The intent is entirely likely to be easy to play, not to be a powerhouse (like the intent of the wild mage is to be wacky to play, not to be a powerhouse).
The second problem is more significant if it exists, but comparing damage between the classes doesn't make that case. D&D isn't numberwang, it's PvE, which means the context that matters is in monster fights, interaction challenges, and environmental challenges, not in simply comparing numberwang.
