With two short rests at level 2-5 expertise is only better if you roll 11 or more checks in the skills you have expertise in. That is not a few, it is a lot.
That is not close at all as I don't remember a single session where we rolled 11 checks in 2 skills in a day.
Right it is a Warrior that blows THE BEST expert out of the water for 20% of the game and is still better than them for another 20% of the game, and I think it stays better than one of the Experts (Ranger) all the way to 20th level.
In terms of class mechanics a Ranger is never a better skill Monkey than a fighter, a Rogue is better for slightly more than half the game if you play all 20 levels, but a Fighter is better at combat for all of the game.
Bard, like other casters, can select spells for the non-combat pillars and be comparable from about level 7 on, and better than a Rogue at high levels too while also being better at combat.
Experts are not the best skill monkeys. A Warrior is at most of tier 1 and 2.
I do not think that Fighters beat out Bards, or many other spellcasters at Tier 2, or even earlier.
Bear in mind that 2 short rests might be what the game was allegedly calibrated around, but it is relatively unusual for 2 short rests to be the norm for a game, with 1 being closer to the average in most it appears.
The two factors you are not taking into into account I believe are:
1: Ability score distribution: Assuming that each character picks skills according to a reasonable understanding of their strengths, a Fighter will have Perception with a +0 ability bonus, Athletics with a +3 ability bonus, and two other skills with probably no, or a small bonus.
A spellcaster might have Perception at a +0 bonus, and three other skills at a +3 ability bonus.
Now, we all know that a fighter could invest on tertiary stats to increase their ability bonus to those, at the expense of combat effectiveness, but we're talking generalities here, and of course, the spellcaster can do the same.
2: Spells. The bonus granted by Tactical mind is quite similar numerically to that granted by Advantage. Even if the spellcaster does not have a spell that will resolve the situation without requiring ability checks, the Enhance Ability spell grants advantage to ability checks as a mere level 2 spell.
Sure the wizard or whatever is burning their combat resource for non-combat situations, but so is the Fighter if they are using Tactical Mind. (I'm not counting Guidance since we generally end up comparing fighter to wizard, but it is definitely a consideration if the spellcaster isn't a wizard.)
Past level 3, the chance that a spellcaster will have a spell that will just be able to resolve the situation increases, but the relative resource cost decreases for them, since they not only gain more level 2 slots, but also higher-level ones as well.