It's not a demerit of 4e that it doesn't support "low-importance" fights as a means of daily attrition. One does not have to like that about the game, but to my mind that's a matter of personal preference, rather than a design problem of the game itself.
At any rate, it's plainly incorrect to say the big set-piece fights don't cause attrition - if each combat runs through three or four of each PC's healing surges, you better believe attrition is happening! And, of course, once the player characters run out of second winds and other special abilities that actually let them use those healing surges, then they can be in real trouble once they start hitting 0 hit points.
Also, the mechanics of 4e just plain clash with large numbers of small grindy fights. Again, that's not a demerit of the game itself, even if no one is obliged to enjoy that element of gameplay. (It might be to the discredit of the game if neither DMG explicitly makes this plain, although if reading the DMG was as common in 4e as it is in 5e, it probably wouldn't matter even if they did.)