I think spell schools are generally a bad way to arrange things as they seem to be pretty arbitrary. I think both Eldritch Knight and Arcane Trickster should have their own bespoke spell lists. Though I understand why they didn't no it, it would use a lot of space.Sounds like a great house-rule.
I've house-ruled the EK several times over the years. Once to remove the school restrictions entirely, and let the EK choose any spells the player watns. Another time, I houseruled it to allow its spells to be chosen from the Bard class list (for a "songblade" flavored fighter), and most recently I've house-ruled it to let the player use spells on the Warlock class list. It works fine.