IIR the goal of "you don't need magic weapons" was more "you don't need any particular magic weapon" rather than "you shouldn't use magic items." Put another way: the game isn't balanced around "there are no magic items" so much as "no specific magic item is required."
I think they were hoping that nonmagic weapon resistance would sometimes matter, but I'm not sure they succeeded. I do know they understood that different tables would play very differently so they tried to make the game robust enough to allow for lots of variation - and they mostly succeeded. The only common variation that flat-out doesn't seem to work is consistently having only one fight per long rest, which does tend to create issues, but for the most part the game does seem to hold up okay under a variety of styles.
The downside of this is that the omni-tool isn't particularly well-designed for any particular task; if you have a style in mind that your whole table prefers there's a better game out there for that specific style.
So, the game does not require that players have +(Tier#-1) magic items in order to be able to defeat level appropriate monsters.
You can compare this to 4e, where if you lacked a level appropriate magic weapon, your wiffs would be insane against a level appropriate foe, especially by epic tier. Losing -6 to hit and -6 AC and -6 to -9 on all of your non-AC defences really crippled you.
Now, in 5e, getting those tiers of items is a
large upgrade. Like equivalent to multiple levels. But as the accuracy bonus is bounded by +0 to +3 it doesn't break the core combat engine; monster ATK and DEF scaling rate is chosen so that the game works without it. (I mean, CR 35 monsters probably need your players to have +2-3 items).
At level 1 you can expect a +4 to hit and 18 AC.
At level 20 with a belt of 27 strength you can reach +17 to hit and 26 AC.
The lowly CR1/8 Guard has an AC of 16 and an attack of +3 is just barely obsolete if you are capped out that way.
Without those magic items, you hit +11 to hit and 20 AC; you still have a chance to miss a chance to be hit by a town guard.
Similarly, spellcasters and melee characters are not eclipsed by the other without magic items. With magic items, well you can make either eclipse the other.
I've talked about inter-melee class balanced. There is currently an itemization problem in 5e whereby the typical magic weapon in published content benefits the Fighter more than other melee types by a large amount. And I pointed out how I get around it.