I worked at a national lab for 5 years and loved it, but I know exactly how you feel about the negative parts. I left and ended up at an ad-tech company and although I didn't like our mission at all, enjoyed the job just because of all the new skills I learned. Eventually I had to leave because I just never could feel good about what we were building.

Then I ended up at my current company. Our mission is to improve supply chains by monitoring vaccines (mostly in Africa at the moment) and perishable goods with our sensor devices. Especially with the vaccines, our mission is literally to save lives. I LOVE it here.

So, the private companies you are looking for are out there, so maybe just stay patient until you find something that you can feel good about.