What was your mixture and trim? Additionally were flaps deployed?
With the C172 I normally start a very shallow climb of 200-600 at the start until I have good speed of around 70-90 knots and then climb. It’s important not to do a steep climb right after takeoff. Remember adjust your VS according to throttle/N1. If you notice your throttle going very high then shallow your climb and find a rate of climb where your throttle is steady.
Yeah I don’t use trim for the 172, also for climb I’d recommend a rich mixture, especially if you are close to stalling.
From what it sounds like, you were climbing too steep with the nose pointed too high from trim and also with a leaner mixture, your engine wasn’t working as hard as it could’ve. I usually use 0 trim or low trim, rich mixture for climb, and as I stated, takeoff and then shallowly climb until you have a higher airspeed and then climb. Also remember to adjust your climb rate to your N1. Hope this helps! :)
You’re roughly 200 pounds from maximum weight. This will limit your rate of climb.
Nevertheless, at maximum weight you still should be able to climb out at 500-700 FPM (depending on the atmospheric conditions) and at 73 IAS. Mixture should be leaned for best performance above 3,000 and flaps retracted.