He may be wrong, and in particular "reasonable doubt" may not have the legal definition that he thinks it does (legal terms rarely match common sense). But his argument is not "Glossip is innocent", but "there is no way we have sufficient evidence to kill this man without risking that we are killing someone who might be innocent".
Have you served on a jury, by the way? I served as the foreman on one relatively minor criminal case, and it did more to shake my faith in the rule-of-law than any other (minimal) encounters I've had with the legal system. In the end, we reached a unanimous "not guilty" in the case.
This was a difficult decision for all of us to reach, since the only way this could be the "correct" decision was if both police officers who testified were lying about significant details both in their report and on the stand. And yet this is the conclusion we eventually all agreed upon as being the most likely explanation for the way the pieces failed to fit together.
The defendant may well have done other illegal things, and possibly the world would be a better place if he was behind bars, but as a jury we decided that there was sufficient reason to doubt that he was guilty as charged. And yet there was no particular guarantee that this case would turn out as it did. On a different day, with a different jury, a different judge, or different lawyers, (but the same defendant, witnesses, and evidence) a guilty verdict seems likely.
As a limited participant in the legal system (but a "logical" computer programmer) my assessment would be that if we have the death penalty, the American jury system is practically guaranteed to occasionally sentence innocent people to death. I'd guess that this is the part that bothers Paul as well, I'd guess that he probably does have experience that leads him to believe this, and I'd guess that his opinion on the matter is not likely to be changed by a counterpoint that makes Glossip's guilt seem somewhat more likely.