If you want a list of labs, here is one from North American Proficiency Testing:
Keep in mind that different labs use various extractants and you cannot compare numbers if lab A uses Mehlich 1 and lab B uses Mehlich 3 (or Bray or Olsen or whatever). Also, you are not sending the exact same soil to different labs. It may be similar but it is not the same sample.
As for Soil Savvy, what you get is a one time recommendation for fertilizer. You won't know your overall shortages, won't get any recommendation for the rest of the year. It does not report cation exhange capacity, which is one of the most important things to know about your soil. There is no buffer test so a meaningful lime recommendation can be done if lime is needed. Also, they even say that the pH they report may be off by 0.5. it's a pretty useless test.
You get more meaningful results if you are consistent. Use a consistent depth, the same lab, same time of year and you can then track changes year to year.