Umm...tell that to the entire feat section. You know, the section littered with feats that allow you to take a -5 penalty to your attack to get some other effect.
Again, look at feats, spells, and even the basic combat rules. There are many many many abilities that push creatures 5-15 feet, and are significantly more beneficial to groups playing with a grid. In fact, because most spells are measured in 5 foot increments, it makes not using a grid very difficult in 5e, unlike say 13th Age which is actually designed around TotM.
Well, except for the two existing fighting styles that already do just that! Archery fighting style is +1 attacks, and defensive fighting style is +1 AC.
How can my proposed suggestion be against the spirit of 5e, if 5e is already doing all of those to a much worse degree!
5E is very hypocritical like that. They had as a design goal allowing everyone to play their own play style, but 5E mainly only covers the 2E/3E play style of system mastery, but not tactically sound choices.
They talk about balance, but fail to improve on the previous editions balance and actually went back a step.
Add in the things you've noted and you get a game that doesn't follow its own design philosophy.
My suggestion is:
Great Weapon Reach
While wielding a two-handed heavy weapon your gain an extended reach of 5 feet. You can attack with your weapon in that range, but you cannot take opportunity attacks at that range.