Ill try to keep the "brutally" in check, but my honest opinion is that with your stats, you will likely land some interviews if you apply broadly. But I really doubt that you would be competitive for HSDM, especially for the joint program. That's not to say you won't get in anywhere, but it's important to manage expectations.
I always try to put myself in others shoes: if I was sitting on the ADCOM and your application came across it would raise my eyebrows. Why? Because you have a relatively competitive GPA, but not for HSDM, and your DAT is not competitive for the program. But if you namedrop someone who is vouching that you are the perfect candidate for one of the most competitive and rigorous programs in the country, it would raise some questions. If they are so competitive, why is the DAT on the lower end of the scale?
We all know admissions is very competitive, and we are all looking for the edge to get in. I agree that you should retake your DAT and aim for a much higher score. If you get a 24+ that LoR will carry more weight because your grades reflect the recommendation and potential for success at that program. It gives the letter more credibility.
For me, I had alum write me a LoR for two different schools that I applied to. I was rejected from one, and accepted to the other. Honestly, I don't think my acceptance was based too heavily on that LoR.
Keep on truckin' man. Keep studying and making your application more competitive. Work on your DAT and continue to improve your application until you get an offer. You'll get something.
Good luck!