Controlling offspring numbers is the first, the easiest and also the most important line of defense against loss of genetic diversity. No one individual should reproduce significantly more than others; creating overcontributing families will lead to more individuals becoming strongly related, which will then lead to unavoidable inbreeding.
Especially when “new blood” (an individual with low [mean kinship]) is introduced, breeders might be tempted to spread the genes of this individual as far as possible - and as quickly as possible - to help with genetic diversity. This will indeed be the case in the first generation, but long-term will always do more harm than good.
The problem of the “popular sire” or dam is two-fold: the gene variants carried by that overcontributing individual will become common in the population, and a couple of generations down the line the latest, the harmful ones will begin to be expressed. At this point it will be too late for breeders to go back, because the options are either to cull a large chunk of the breed or just continue with the new or increased problems. Additionally, having one individual breed extensively means other potential breeding animals will be used less, or not at all, effectively deleting their unique genetic diversity from the breed.
In a small breed one individual should not contribute more than 5% of dogs of a generation
[1].
Offspring numbers should be tracked in second and third generation as well as in the first. Second generation contribution should stay under 10% and third generation contribution under 20%. For animals with high contributing parents or grandparents, and litters with a high combined contribution between several siblings the situation should be even more carefully monitored.