Here's how I look at it. In theory, consistent neck tension can improve precision and that's a good reason to anneal every cycle. However, Brian Litz didn't detect any difference between anneal-every-time and cases which were shot ten times without annealing. It was a careful but limited test, so take it for what it's worth.
If there is an improvement, it isn't easy to prove it, but like many tiny steps which might help (neck turning, chamfering, pocket uniforming, trimming, etc), I do it because I think they all add up to a significant difference in precision even though proving any one of them is difficult without wearing out your barrel.
Keep in mind that "shooting good groups without annealing" can mean hitting a paper plate at 100 yards half the time or shooting bug holes nearly all the time. This is the Internet, so take what you read (including this post) for what it's worth.
One thing I know for sure is that I used to experience split neck rather quickly before I started annealing. Now I don't keep track of how many times I fire my brass and I simply don't consider split necks; they don't happen anymore. My brass seems to last forever, or at least until the primer pockets gets loose which is nearly forever. Incidentally, most shooters I compete against use Lapua brass and I do too. It's more expensive but it is well made and durable to the point that my brass cost per round fired is so low that I don't even consider it.
Annealing can be tedious and you can also spend a lot of money on a machine or both. I solved both problems by building my own for a little over a hundred bucks. It has an auto feed chute, it's accurate, adjustable, doesn't overheat, and it even has a round counter. I anneal my three competition calibers, .223, 6mm BR Norma, and 6.5 Creedmoor. Changing calibers takes literally ten seconds by adding a small shim.
It is so easy to use, that annealing isn't tedious so there is no reason not to do it every cycle. I seat bullets with a Wilson die and an arbor press which allows me to feel the seating force even though I can't actually measure it. Annealing every time lets me make 70 rounds of F-Class ammo where usually 68 or so feel identical and the two odd balls wind up as spare sighters. Annealing every cycle helps me achieve that consistency which may help at the target, or at least is unlikely to hurt.