Welcome JJ,
I can hear the frustration from the delay in finding the cancer. I would suspect that you have dense breast tissue. Did you ever have an ultrasound or MRI?
The best advice I can give to you is to be more proactive on seeking 2nd opinions. Physicians are human, subject to human error and limitations. It is amazing what you can learn on a 2nd or even a 3rd opinion. Right now I'm looking at the comment your oncologist made that your treatment only "guarantees" success for 3-5 years and I am not personally comfortable with that statement. There are no guarantees with cancer. If you check the American Cancer Society website the 5 year survival rate for stage IIIa is 67%. But that lumps everyone in, including luminal Bs & triple negatives who are more likely to have a recurrence. You should do well despite the delay in treatment. I personally expect that you'll be around for decades
With the paclitaxel you might try icing your hands and feet during chemo to help ward off neuropathy. I received B12 injections to help with this issue.
Penny is taking Biotin to help avoid the nail issue. You might want to give that a try.
Keep us posting on your progress. Sounds like you had a good plan with the chemo, radiation and reconstruction.