This guide really undersells War Magic with the new melee cantrips.
At 7-10, you're basically getting both of your normal attacks, plus add the effects of the cantrip (at least 1d8 damage, plus boom or cleave damage).
At 11-15, you trade one attack (avg ~13 damage for great weapon style) to get 2d8 (average 9 damage plus cleave or boom effect). A worthy trade in almost all situations, IMO.
16-19 trade one attack for 3d8 (13.5 avg) + bonus effects. Simply a no-brainer.
Only at level 20 does four attacks definitively pull ahead of War Magic, but only really for a single target whose movement you don't want to control.
War Magic also has crazy synergy with Sorcerer and Warlock multiclass, which is very viable because EKs can easily dump Int and grab buff spells to focus on Cha as an attack stat.
A level 11 dex fighter gets 3 bow attacks for 1d8 + Dex, while an EK/warlock multiclass gets 3 attacks for 1d10 + Cha plus 1 bow attack for 1d8 + Dex, each with 1d6 bonus damage if Hex is up. It's probably the most dangerous ranged attacker short of Swift Quiver plus Sharpshooter, and it doesn't cost any feats or high level spell slots.
An EK/Sorcerer multiclass can twin Booming Blade + War Magic to get just as many attacks as a straight fighter, but with a bonus 4d8 damage plus boom effects. Requires two adjacent foes, but it does great damage and controls two foes!
I'd also quibble with some of your spell recommendations too, but I think I've done enough damage for one post.