My husband usually bicycles home from giving blood, and doesn’t notice any recovery needed, beyond what they make him do there. When I used to give blood I always felt tired for a couple days. I also used it as an excuse to make chicken livers wrapped in bacon (which I love) since liver is very high in the easiest-to-absorb type of iron.
It does take some time to refresh your red blood cells. If you have more than you need, you won’t notice anything. But it doesn’t take a lot of time to refresh them. When I visited Utah I found it significantly easier to hike at high altitude on day 5 than on day 1. Pretty sure that was because my body reacted to spending time in thin air.
Given that some athletes “dope” by getting an infusion of their own red blood cells, it seems plausible that a runner would compete less effectively right after donating blood. But red blood cells only live 120 days, and I’ve felt my red cells boost from being “low” for high altitude, and i also recovered in a couple of weeks after i lost a lot of blood to a breach delivery. i bet they regenerate after donation, as well.
Bouncing back in a day? Probably not, at least for top aerobic performance. In a couple of weeks? Yeah, I’d expect that.