Whether or not it would have been a good idea to loosen the approval process aside, accelerating it would likely not have had much of any effect on saving those

Whether or not it would have been a good idea to loosen the approval process aside, accelerating it would likely not have had much of any effect on saving those lives. The bottleneck is production capacity. As I understand it Moderna and Pfizer/Biontech have both been building production capacity and creating vaccines as fast as possible, but have only now created enough to vaccinate a few tens of millions of people combined. If they had been approved in the summer, when they went to Phase III trials, they might have, perhaps, been able to turn out a few thousands vials a day which is not really enough to meaningfully impact the spread or death rates even if you were able to highly target high-risk groups.