I have been a troop leader since my daughter was a Daisy in kindergarten. She is now in the fifth grade. We have a large troop and there are three of us who are co-leaders. From the very beginning, we have encouraged moms to do what was comfortable for them. We have our meetings directly after school, at the school, so there is no dropping off, but there is a specified pick-up time. Whenever a mom expresses interest in helping we welcome her to a meeting. However, they are allowed one freebie. After that, as mandated by the rules of The Girl Scouts, they have to register as an Adult Volunteer and pay a fee. It is due to the insurance rules as explained before. The way the world is today, there are all kinds of requirements we must abide: training, girl-to-leader ratios, etc. When we went to a Girl Scout camporee last year, none of the leaders/volunteers could sleep in the same tents with their girls. Just one of the rules. It's very strict.
Sometimes a mom (or dad) has a specific skill or career that is germain to the badge we are working on and they come in and do a lesson. For example, we've had a dad who is a policeman talk about safety, a mom who knows sign language come in and teach during our discussion on different methods of communication, a mom who had a great idea for a craft come in and run that part of the meeting, and so on. They usually had something specific to contribute.
But generally speaking, very seldom do we have moms want to stay, except when we have a field trip and Moms are needed for car pooling. Then, luckily,we have an abundance of volunteers--again they must be registered.
I have found that it is much easier to keep the girls together as a group when their own parents aren't there. Even my own daughter, who is extremely well-behaved, can sometimes try to take advantage of having her mom around. It is kind of like being in school--you have a schedule, and activities and everyone needs to follow the same rules and they need to know who the authority figures are. It can get tough with a lot of moms around.
I wouldn't be too tough on the moms who "drop and run". I don't think they necessarily see it as "free babysitting", but rather a time when they can do their grocery shopping or tend to another child while their daughter is occupied. My daughter is on two premier sports teams, and I don't look at her practice time as "free babysitting". In fact sometimes the schedule is just a pain I have to deal with while I have to get my other stuff done. And similarly, the coach will let you stay at these practices, but they would rather you didn't. They just don't want the kids self-conscious or concentrating on their parents when they should be paying attention to the coach. And they certainly don't want the parents yelling out instructions to the players. Again the coaches/teacher/troop leaders need to be seen as authority figures who are running the show.
I have also found that most kids don't want their parents around--doing something on their own gives them a sense of independence. I really think that if you aren't comfortable leaving your daughter for the meetings maybe you and your daughter aren't ready for this kind of commitment yet. After all, she is little and maybe you just aren't there yet. Or if you aren't comfortable with the leader in general, I think that that is a valid concern and you could try to get into another troop, or you could try to become a co-leader, or offer your services for whenever the leader has a complicated meeting or a messy craft.
As far as this leader is concerned, like others have said, maybe she just doesn't communicate properly or is a little overwhelmed and having parents "watch" her makes her nervous. My advice is to get to know her better, and ask her how you can help her without getting in her way. To get to this point she has gone through the background checks and done the training. Maybe she just needs a little more experience.