I've used to use an app (there are many) which measures angular velocity using accelerometer and displays RPM (you place the phone on the turntable). I can't see why would anyone need something other than that.
Complete noob here, but wouldn't the phones weight (especially with how big phones are nowadays) have a significant impact on the RPM by slowing it down?
It's very possible, increased weight will increase friction in the bearing. However my iphone SE weighs almost the same as a 12-inch record (150g). BUT it is also more difficult to place symmetrically on the turntable, which also could affect friction.