Pullups are a good functional movement. You don’t want to arch your spine, but remember to keep your core engaged, your hips aligned, and keep your elbows in as you pullup.
There are other variations like the rings or TRX. With the rings, your feet will stay on the ground, keep your core tight, and you’ll start with knuckles up and rotate them 90 degrees as you pullup.
Buildup your pullups, put the work in, put the time in, and you will see the progressions that you want, and it will help your overall performance.