Hey, no problem. Support did explain the reasoning behind our initial issue. It had to deal with the mapped tape plan always needing to run a virtual full if no active full is scheduled, which is what we caused by unchecking the Friday Weekly full disk backup. They basically said we could continue to uncheck the Friday Weekly full disk backup whenever we run a monthly on that Friday instead, but we should then just "trick" the Veeam scheduling, by changing the day the virtual full backup is set to run, then changing it back, since the virtual full backup setting cannot be turned off. I think this should suffice for now.
I am not sure if I have a good answer for your question of why we are using the "create new media set..." setting, except that, and correct me if I'm wrong, it seems much cleaner to do so. It is nice to know each Friday Weekly full is on one tape only, and is completed during that Friday run. From my understanding, if we append to the tapes instead, while doing a Friday Weekly full, that tape could get full, and require a second tape to finish the backup, which we couldn't do until the following Monday since we only have a single tape drive, and no one is available to change tapes on weekends. Thus we actually wouldn't have the most recent full backup until the next Monday, instead of completing on that same last Friday. It seems there is just too much manual tape changing required.
Please feel free to educate me on my inaccuracies. Thanks for all the help.