I like cornpone's answer, the tell is in the tape.
A chronograph is a valuable tool, and we all knew what you meant. It will give you the velocity, a number that is needed to calculate fpe, FT/LB, FT-LB,,,,,,,,,,,,,aka, energy. Don't overthink "energy", it is a very confusing term to some. It is the force that moves things, drives things, and breaks things.
Enough is needed, but too much isn't going to be a lot of negative, especially in an air gun. You will not reach the speeds to turn a temporary cavity into a permanent cavity, so think, the ammo has to touch, tear, or crush vitals in order to do damage.
Shopping for the best performing pellet/slug for your purpose, the velocity is valuable. Different compositions do different things on the terminal end. Shooting a 750 fps performing round isn't going to do you a lot of good at 650 fps. MV is a great way to measure certain things, but impact velocity can be calculated with a starting point. I'm probably over explaining. If you are shooting paper, all this is kind of overkill. If you're happy with groups, running it across one at a range or with a friend who has one will give you some numbers to keep in mind without having to pony up for a device. FWIW, most are very accurate these days and the measurable difference in properly working devices, be it $100 or $900 is negligible. Features are what you are usually paying for.
NEED? No you really don't. If you are trying to get the most velocity, set it that way, then fine tune for group size. Even when loading powder burning ammo, I'll get to an accuracy node in the velocity range I'm looking for and load 1/10th grain difference to fine tune. At that point, I don't usually bother with the chrono, but we like to obsess and overthink things as a group.
Happy Shooting