Not wrong. You are playing correctly for you except that you aren't happy with it.
I have never cleared 2 POI's on my first day. I usually take over the first one I come to which is usually one of the trashy non-level designed wilderness houses. I improve the upper floor and remove access to it, finish up my quest by placing a fireplace, replacing my bedroll, and crafting a chest to empty everything into. Then I spend the rest of the day gathering basic materials.
Day 2 I travel to the trader and often skip zombies I encounter in the wilderness. I hunt every animal I come across to the trader and back again to my initial camp. Then I start exploring outward and scavenging in POI's. I do not hunt down and kill every zombie. I let wandering hordes move past. When threatened I kill to defend. I also start working on making flagstone blocks in anticipation of my first base.
I usually stay in the first house until after Day 7 which pretty much hoses it. Then I move closer to the trader and build my own base. Week 2 is getting that base ready. I am usually level 8-10 by Day 7 and having a great time. I have zero complaints about my pacing or what the game throws at me. The game plays out like an apocalyptic story or an episode of The Walking Dead. But playing like this....letting zombies go by....would probably drive you crazy, eh?