I read about 10 different guides, most trying this with a single settler and almost everyone omits this one important detail:
-> always sleep for 48h in a bed (2x 24h)
doing this somehow accelerates how fast happiness goes up... sleeping for 24h it'll go up in about 20 minutes, while sleeping for 48h will up the happiness in 4 minutes. If the happiness doesn't go any further after 5min, don't even bother waiting any longer... sleep for another 48h, rinse and repeat.
Also, you have to be around. Settlers are happier when you're nearby, so don't even think about leaving the settlement.
Most guides will take dozens of hours in real-life time to get to 100. That is, if it doesn't get stuck in 90, 93 or so on. So, I personally did the 6 settlers format w/ 7 beds (1 for each and 1 for me) in Red Rocket Truck Stop. 5 settlers in lvl5 clinics, and 1 taking care of 6 mutfruit trees, w/ 3 water pumps and 70defense.
With this method, I got the achievement in less than a real-life hour .
While I waited, I jammed paintings in every single wall I could, both to pass the time while boosting happiness, and to increase the settlement size (the achievement requires it to be "large", thus the size bar has to get yellow).