Dry oaks are always a gamble; they can be rotten on the inside, even if they look solid on the outside. I'd definitely avoid doing it myself. Typically, reputable professionals come to assess the scope of the job and the risks first, and only then quote a price. If they offer to do it "by eye" for pennies, pass.
Well, yes, in the situation you described, there is a risk if the team doesn't have proper safety equipment and proper safety equipment. From my experience, I can tell you: you can't just drop heavy branches; precise cutting and hanging techniques are required. I used tree removal palo alto for similar tasks. Check out their website for more details. In my case, I really liked that they're a licensed company, their workers are fully insured, and they have all the necessary equipment for cutting and chipping wood. They come out for a free estimate, fix the price immediately, and clean up the site completely after the felling. Check their website; it's much more reliable than hiring undocumented individuals.