I probably have one of the best second race improvers of all time....
Guinea Saurus Rex. Finished 7th in his MSW debut with a 53 speed figure. Next out won with an 81 speed figure. Went on to win over $2 million (at a time when that was a pretty big amount for a turf miler) including 2 Steward's Cup runner up performances. But the fact that I'm talking about a horse from year 36 should give you a feel for how common its been in my experience.
I breed between 40-50 horses/year and race them all. I can't think of any instances where I've had a solid/claimer galloper go on to be a stakes winner, much less in their second start show that type of talent. I definitely see horses improve with time fairly often but it usually takes a bit and its usually a one level improvement, so ok stakes horses can all of a sudden become super stars, allowance type horses start winning stakes, claimers move up to allowances, etc. I do sometimes see horses improve noticeably start 2 but I feel its most commonly a horse that seemed to underperform to their gallop in their first start. I did have one productive galloper I can think of become a stakes winner,
Seven (coincidentally by Guinea Saurus Rex) but he actually ran quiet well first out and continued galloping productive while winning stakes and then with age started actually running like a productive galloper.
So based on my experiences I definitely think there is an argument to race all your horses, you can often earn more money in claimers than you can selling a subpar galloper and certainly lots of horses will show some level of improvement with time, but if you are racing all your horses expecting your claimers to one day become super stars you will be disappointed more often than not.
Diclaimer to all this, I don't race any routers which I get a feeling from things the steward has said in the past is where there is the most potential for improvement as they are not meant to be precocious horses.