The way the dice roll even slightly will probably be a much bigger driver than +1 weapons
This doesn't really make sense to me. Unless the dice are biased, over time they will tend towards the expected average. With a +1 weapon that expected average is higher.
1 pip on a d20 = 5%.....
so roughly a
Common magic item +1 = +5% to your die roll..
Worse case... a +2 bonus = a +10% to your die roll.
Not a huge mathematical variance
+1 to hit and damage? They increase the damage of one party member by about 8% or so....the math is more complicated than that, but it's a rough estimate.
Your rough estimate is wrong. It's about 20% for a +1 weapon (maths below).
It's not though just one +5% on to hit rolls, it's other bonuses too. Damage rolls, less affected by creatures that resist non-magical attacks, higher AC form magic armour/shields, and I imagien a variety of other effects - D&D magic has always been good at that. And incidentally, I dispute that it's a 5% increase in accuracy. An increase from hitting 50% of the time to hitting 55% of the time mean you hit 10% more often - 50 to 55 - and as the target number increases each point on your bonus becomes more valuable.
In the playtest, a lot of creatures like orcs, goblins, warriors, etc seem to have ACs from 12 to 16. To hit, at 1st level, is +2 from proficiency +3 (let's say) from strength. For +5 overall, or a chance to hit between 70% and 50%. A +1 is therefore +1/14 to +1/10 increase in hits.
Damage seems to be 1d8+3 or 1d12+3 (the bonus is from STR). That's an average of 7.5 or 9.5. With a +1 it becomes 8.5 or 10.5.
The overall expected output vs AC 12 goes from 5.25 (d8 weapon) or 6.65 (d12 weapon) to 6.375 (ie more than +20%) or 7.875 (ie a bit over 18%).
The overall expected output vs AC 16 goes from 3.75 (d8 weapon) or 4.75 (d12 weapon) to 4.675 (nearly 25%) or 5.775 (well over 20%).
I think that increases in damage output of around 20% per attack will be noticeable in play.
That's all just presentation. "100% percent more" makes it sound like a lot, but it's still just a mere 20th of your rolls. Not a big deal.
You're still playing games with ratios. Yes, 1/10 is double the output of 1/20. It's still 1/10.
Of all your d20 rolls, 1 in 10 will be a success that would have been a failure without the item.
And if you're dealing an average of 10 damage on a hit, that's another +1 to damage. Before we actually apply the damage bonus from the weapon. I don't think that will be invisible in play.
Both 3E and 4e contain feats that grant no bonus to hit but a +2 bonus to damage. Those sorts of feats are fairly popular, because they give a noticeable increase in character damage output: around +25% for a character with 16 STR wielding a longsword.
More generally - it can't be true
both that magic weapons give you an advantage in play, and that they make no difference in play.
EDIT: Following on from [MENTION=49017]Bluenose[/MENTION]'s post above, the higher the other bonuses on the character the less a +1 weapon will contribute - eg prof gains, stat gains etc. But "fewer buffs" is meant to be a catchcry of the edition; and at higher level typical enemy ACs will increase at least a bit, I think, soaking up some of those prof gains.