If I might butt in here, speaking as a newer admin, you are both sort of correct. In the case of AFDs, the closing admin is expected to analyze the debate and either keep or delete the article based on its content. Speedy deletions, however, can be carried out by an admin without any community input. This is because speedy deletions are only for unequivocal cases where there is no need for a debate. The point is, these supposed "privileges" come with a lot of extra baggage. Admins need to carefully consider the consequences of their admin actions before making them. Even if they make the right decision, they often have to deal with angry users who are upset because they believe the admin acted incorrectly. That is why we have WP:RFA, to weed out those individuals that lack the judgement, restraint, and policy knowledge needed to wield the extra tools.