Over all at 20th level action surge and fighting style balance out the loss of +4 strength. Using second wind covers about 65% of the hp loss from +4 Con (you are still 2 points lower on your con save). That leave the full class barbarian with Indomitable Might, unlimited rages (I'm used to only 6 combats per day so this to me is a moot point) and an ASI, compared to one tool proficiency, 4 battle dice d8, and 3 maneuvers. Over all i feel that they are balanced, dipping fighter gives you a little more damage while staying barbarian gives you more surviveability. The main benefit I find from dipping fighter is that you do not have to wait until level 20 to get all this extra damage and health, instead you get it at a much earlier level so it is more applicable over the coarse of your characters life.
You are clearly ignoring some points. +4 strength is +2 to hit and damage. The +2 to hit makes the use of Great Weapon Mastery (-5 to hit, +10 damage) even more appealing, especially with Recklessness. Those 4 d8 superiority die (each only fuels a single damage roll) totally lose out on +12 damage (more strength + GWM) on each successful attack.
With +4 Con (24 total) you can achieve 7 temp HP per round (as long as you are being reckless), instead of 5 and a meager d10+3 (average 9) per rest. Also Second Wind requires a bonus action to trigger, so one less extra attack with spiked armor, which will hurt your damage output. There goes a d4+7, or even d12+17 if you manage to kill/critically hit a creature during your turn (GWM). You have a total of 285 HP and lvl 20 instead of 242 HP (3 lvls of fighter, otherwise it would be 245) and you have it easier to succeed on Relentless Rage a second, a third or even a fourth time, especially since you saved one more ASI/feat (lvl19) to choose Lucky. When it comes to resting it's 20x d12+7 vs 17 x d12+5 + 3x d10+5.
And infinite rage also means infinite Feral Instinct.
Of course this will all be redundant if you quit playing your character after reaching lvl 20. If your character continues on some kind of epic campaign (you can still acquire items instead of lvls), you should take this into consideration.
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