I don't think your idea is game breaking. You are just house-ruling that once they have accumulated a total of 5 martial classes, they get Extra Attack. Since there are a lot of goodies to be had from 1 level dips already, I would suggest one change:
Allow the player to trade your feature, which I will call "Additional Martial Training" for one of the other abilities gained at the first class level. For example, upon dipping into a level of fighter they could choose to trade a fighting style or Second Wind for Additional Martial Training. You could even slice it thinner than that, letting Second Wind recharge on a LR instead of a SR. Maybe a dip into Barbarian would allow trading Additional Martial Training for one rage/day, +2 rage damage, or Unarmored Defense.
My first character using your rules would be a strength-based rogue: Barbarian 2/Fighter 3/Rogue X. With Extra Attack, reckless sneak attack with my rapier, Dueling FS, Action Surge, Danger Sense, Superiority Dice (including Riposte to have a chance to land a second reckless sneak attack), and Cunning Action I'm pretty confident that I could be a nightmare in combat. Extra Attack opens the door to using the first attack to shove a creature prone, which is almost guaranteed to succeed with Athletics Expertise and advantage while raging, and then landing my sneak attack in the same turn, while giving my allies advantage as well. I won't ever have to worry about allies being near an enemy in order to sneak attack because I can give myself advantage at will. If I knock them prone, I can sneak attack and stay put, hoping to use riposte to trigger a second sneak attack with my reaction. If I don't knock them prone, I use reckless attack for my sneak attack and then use Cunning Action to disengage.