My daughter was much like yours at that age - February birthday and all. She started at two and a half, in the fall before her third birthday, but her preschool actually had a class for kids this age so it wasn't like your situation where they're making an exception.
If she is ready to go, you know it. I think she will be happy. I would not worry too much about the idea of "she'll think she gets a second chance to do the same class" because she still will be just too young to think that way; and remember, to her (and I hope to you too) it is not a "class" in the sense of academic achievement-- preschool is for socialization and learning to get along in a group, especially between about two and four years old. She also will bounce back just fine from her friends going to another class, especially if you make the effort to have some play dates outside preschool when she starts the second year with a new group of kids.
The one thing I'd want to check into: Is this preschool...twice a week for three hours? Five days a week for five hours? "Preschools" can have vastly different schedules. I hope you're talking about something more like two to three days a week at the most, for just the morning or just the afternoon. At her age she does still need down time, even if she has given up napping, and she needs alone time when she is not in a group of other kids -- going from being at home full-time to being in a classroom of kids five days a week all day long at age three would be far too much, too fast.
Our preschool, and many others, gradually worked kids up in time. They went only one day a week for three hours the first two weeks, then two days a week for three hours, then three days, and that was the maximum for that first year. It was a good system. By the year before kindergarten my daughter went four mornings a week but not all day, every day.
I do totally get where you're coming from. My daughter also was advanced at that age and a great talker and very eager to go to "school." Just remember as she goes through preschool that she's still very young and needs the down time, and the one-on-one play dates, that preschool doesn't provide. But I would send her -- unless it's a very long schedule.
And yes, as someone else posted, I'd nicely ask the school what changed their minds about admitting her; I assume it's because a slot opened up but it would be nice to know.