ABSITE score cutoffs correlate reasonably well with passage of the Qualifying exam (written portion) for the American Board of Surgery. Don't have the data at my fingertips, but I think that the cutoff is ~30%ile give or take. Below that puts you at serious risk of non-passage.
With that in mind, I think that ABSITEs and Board Exams probably are not a great indicator of how "good" a surgeon you might be or become. They test your ability to come up with the "board answer" which as anyone can attest to is often only 1 of many ways to tackle a problem. Each person is an individual and those with only one answer in mind will mistreat some patients by following the book. Having said that, the board answer typically represents either a "safe answer" or a consensus answer and shouldn't be tossed away lightly. When in doubt, the board answer will generally at least steer you in the right direction.
Getting back to the original poster's question: do ABSITE scores predict the quality of didactics? I would say on average, no. I have come across a few programs (primarily community programs in my limited experience) have have intensive ABSITE review plans for their residents that do pay off in extraordinarily high scores, but those are the exception rather than the rule. Ironically, amongst my friends, we often did the worst on the stuff that we knew the best (the future oncologist getting cancer questions wrong, future CT surgeon getting cardiac questions wrong, etc.) because we rebelled against the "board answer" because we had learned way too much about a specific topic to be able to ever answer "C."
To evaluate didactics, I would look for a few things.
Availability: Some type of structured plan for covering all the topics over the course of a reasonable amount of time (e.g. 2 years if not atypical). A mock oral exam for at least the more senior residents. A reasonable schedule of case conferences where more difficult cases are discussed. Ideally, these would be a mix of multi-disciplinary conferences such as tumor board or endovascular conference, and surgeon only conferences.
Emphasis on attendance: How often can residents get to their conferences? Is there any protection/expectation that conference comes first in the priority list (besides coding patients, etc.)? Be wary of programs that residents say they typically don't get to conference much and that they are happy about that. It indicates either a bad conference so they are avoiding it, or overwork on the part of the residents.
ABSITE scores should really only be a major issue in my opinion if the program consistently has very poor scores (<25%ile) among multiple residents of multiple levels and/or board passage rates are low (nationally ~75% of those taking the writtens pass on the 1st try, ideally your program does better than that). The other time it should be an issue is if they have a policy such as all those <50%ile are on probation, and then don't provide you the resources to keep your scores up in the form of good didactics.