Not at low levels which is where it matters more.It's campaign/DM dependent, I think. If the enemies are intelligent, they'll target the caster in the back, so having better self-defense serves your support role. Plus, if you don't have to use cutting words for yourself, you can be handing out more inspiration dice, which for a valor bard are the same dice as their bonus option, so they become more versatile.
Even with magic initiate for mage armor and maxed DEX, the lore bard can get to 18. If the valor bard takes the feat you spent for mage armor and takes medium armor master, they would have a 20, without putting anything into DEX after chargen.
You can max your Dex trade off is your charisma.
If you do that you're not getting much out of gwf.
That's the problem all these gish classes have. You have medium armor, you need a 14 Dex. You have a d8 hit dice so you kind of want 14 con.
Then you kind of want a half decent spellcasting stat and unlike a wizard a bard kind of wants a 14 Cha minimum .
But if you want to be good with weapons you kind if want 16+ Dex or strength. If you have high Dex it makes the medium armor part semi pointless so you're not really any better off than the lore bard in light armor since the other one is Dex based.
If you take medium armor master your AC is ok but you're not taking gwm and have to be a variant human or give up an asi latter.
Half elves can get higher stats, no level 1 feat.
Once all that is done you're waiting a while to do much level 10 or 14 until then you're a sub par fighter type.
Or you can keep it simple and just be a great support character and spellcaster.
It's probably why they invented the hexblade. Wear medium armor and key everything off charisma.