Ugh I’m sorry. I feel for you and really relate. We have an [name_f]Eliana[/name_f] nn [name_f]Ellie[/name_f]- very common when you account for all the actual and nickname Ellies 
And it also seems to be worse in the state we moved to.
I’d give it more time and try not to read into your initial experiences too much. That can’t be representative of the actual ratio. We had an experience like that after we moved (an [name_f]Ellie[/name_f], [name_u]Elia[/name_u], and our [name_f]Eliana[/name_f] all on the same small soccer team) - it was pretty ridiculous but nothing that extreme has happened since.
[name_f]My[/name_f] advice is to try and embrace [name_f]Lily[/name_f] and reconnect with what drew you to it in the first place 
It’s a beautiful name, it’s what your daughter knows herself as, and the popularity is an affirmation of how great it is.
Changing what you call her sounds like a drastic option, and it might not play out the way you hope. There’s no guarantee that there will be another [name_f]Lily[/name_f] in her grade at school, or that she’ll be the only [name_u]Alex[/name_u] (or anything else) in class. Would that make you regret the switch? It seems a big sacrifice to give up a name you like, and you could still run into the same problem anyway.
As a playground solution, I like others’ suggestions of using her middle name or some variation to distinguish her when necessary: maybe [name_f]Lily[/name_f] [name_u]Alex[/name_u], [name_f]Lily[/name_f] [name_u]Lex[/name_u], [name_f]Lily[/name_f] [name_u]Xan[/name_u], [name_f]Lily[/name_f] X. Then you get to keep her beautiful name and she can still feel unique 