I was under the impression an S1 was a similar outcome to the generation of seeds the mother came from.
Depend of your mother. She could be anything, from a polyhybrid to an inbred line. So the result of a S1 will be pretty different. You say that because again it work genetically like a F2.
For example if you made a F1 cross, to make F2 you can take two very different parents, in that case making F2 will reveal a lot of new phenotype (and probably dominant trait) and won't be very different from F1, especially if you cross two line that are highly heterozygous. But if you want to stabilize you will take a male and female that are the closest you can find. In that case, that you take a male that is close to the female, or that you selfed the female, it's pretty the same.
But then in your S1 or F2 you will get more diversity, because the recessive pheno that show up, or no deiversity if your parent line is pure. In the first case you will have to go to F3 to have more stability again, and it's all in the choice of parents.
When Elmer Bud said S1 are great to see if a line is stable, indeed. If you don't get much recessive pheno then yes the line is "stable", homozygous. But if a line is not you will find plant that are "purified" genetically from trait you don't want, but to try them you will have to S1 again to see if this S1 is not also unstable. For exemple let's say you have Aa in your mother, A Is dominant. By selfing you will get AA, Aa and aa. AA will be of course the one to keep for breeding if you want that dominant trait to pass to the progeny.
Also to answer to the question of the topic, which is were we can go by selfing. As i said when you selfed you still get individual with pretty much of the genetics of the parents and other that lost a lot this material, so if your line if heterozygous and your selection is good you can go pretty far. In a way S1 are nt so much destructive that go though classic inbred, it's just quicker to get what you want if you can make proper selection. But as in inbred line, except if you can grow a lot of plant to find that rare perfect specimen, you will probably decrease vigor pretty fast also, but it's the same problem with inbreeding.