Totally agree. In our jurisdiction, the sewer stopped outside the area in which our home is located and never was expanded. Of course, the low-lying area issue that would have required a few thousand dollars extra 50 years ago now would be a major expense. The low-lying areas are now, of course, a major environmental concern, making replacing low-lying septic systems even more expensive. Logically, those should have been the first, not the last, to be connected to the sewer, but municipal projects are born of municipal politics, so...
Obviously, you are more informed than most people asking such questions here. A couple of higher-level thoughts, then.
One issue for you will be that "Gray Water" in the New York Plumbing Code is defined as
"Waste water discharged from lavatories, bathtubs, showers, clothes
washers and laundry sinks." The reason for this is that stuff from the dishwasher has food particles that really need to be processed by your septic system. (This definition is only for limited approved greywater treatment and recycling systems, so even if you had an approved filtration system that was recycling your greywater for use to flush toilets, for example, you couldn't run the dishwasher into it.)
That drywell could get really nasty with grease and food particles flowing into it.
Beyond that, though, Appendix 75-A of the New York Wastewater Treatment Standards promulgated by the NYS Department of Health (which cannot be relaxed by local ordinance), defines "sewage" to mean "the combination of human and household waste with water which is discharged to the home plumbing system including the waste from a flush toilet, bath, sink, lavatory, dishwashing or laundry machine, or the water-carried waste from any other fixture, equipment or machine". And "sewage" is supposed to go into the septic system, with specific exceptions that don't include the dishwasher.
Given that your system was probably designed for 150 gallons/bedroom/day, that six gallons isn't going to hurt the thing. And the flow in the lines might help push along organic matter that isn't getting washed out by toilets and showers the way it used to be with higher-flow fixtures.
Good luck!