They were the third At-Large team, closer to the fourth team than they were to the second team.
I think they proved that they belonged in the conversation, but even with probably the best race of their season, they were still among the bottom half of the teams included (at-large or otherwise). That's not to say that other teams that weren't invited wouldn't have done any better or worse, because they weren't in the field. Saratoga's resume and NXR performance was still weaker, and we don't know if other teams would have done better or worse if they were given the chance.
What was proven, if anything, was that if a team runs well at NXN, they can move up the field more than some might believe - because almost all of the teams that make NXN are close to if not among the top 20-25 in the nation.
Saratoga ran a GREAT race at NXN. They should be proud and should be commended for that accomplishment - but it says and changes nothing about what happened on the way to NXN, which is what people had a problem with (that they really weren't all that good during the season or at NXR, so they hadn't really done anything to deserve their inclusion over other potential At-Large teams that had done more either at NXR or during the season).