OP, I am VERY sympathetic to your issue with labels.
But 2 things I want to say:
1) As the mother of a boy who was mislabeled and HARMED by that mislabel, I totally get it.
Just some back story - as I mentioned DS has a severe language disorder, receptive and expressive. Which means he, unlike your DD, does not understand much of what is said to him. This means that for a long time he didn't respond to his name, often ignored people, seemed off in his own world when not around people he knew, etc. A lot of red flags for autism. He was unofficially labeled as being on the spectrum by every specialist that worked with him and they treated him as such. So he wasn't officially labeled but for all practical purposes was given an ASD diagnosis.
The harm was done by the methods that virtually every therapist and expert we dealt with applied to his treatment. They all used what is called ABA methodology, a behavioral approach developed for children on the spectrum. Because most kids on the spectrum lack the will to communicate and are not motivated by social constraints, ABA uses drilling and a very simple reward/punishment structure. They sit kids at a table and DO force them to drill. This can be very effective and helpful for children on the spectrum but was DISASTROUS for DS. He withdrew emotionally and started tantruming often. These tantrums were seen by the experts as further proof of his inability to "deal with stress or participate in things he does not want to." Bad cycle! That went on for about 3 months and I regret those months with every fiber of my being.
It was when he was about 2.5 that I began to seriously question his diagnosis and got on the waiting list for a development pediatrician. Around that time, the psychologist at the special ed preschool we had DS in called me in to ask about his diagnosis. She also doubted that he was on the spectrum which gave me some confidence in my own observations and I pulled him out of the ABA based school. His tantrums literally stopped the day I told him we weren't going back to school.
We saw the Dr when DS was almost 3 and she was initially confused by his mix of ASD-like behaviors coupled with many things incompatible with an ASD diagnosis. She saw him 4 times in a variety of environments and said by the end that he was 100% not on the spectrum and, instead, has a language disorder (which can easily be mistaken for ASD by those not used to looking for anything else). DH and I completely agree with this as it explains all his behaviors!
My point being that there CAN be down sides to evaluations and early intervention. There is often a rush to label that really can be problematic.
2) That all said, I now know that the kinds of therapy he was getting is NOT what most kids with language issues should receive. In fact, before 3 or 4 years old, all children should be getting what is called play-based, child-led therapy. This kind of therapy (also often called floortime) is literally like playing for the kids. The EI in our area is just kind of caught in the 70s when it comes to using ABA therapy ofr young children. I feel like your impression of therapy in general is very "ABA-ish" and that is not at all what DS gets now.
DS now has a speech therapist come to our house twice a week and she literally just plays with him. But in ways they help him talk more. He LOVES when she comes and looks forward to it. She also spends a lot of time helping me learn and practice techniques that I can use in our every day lives. Because honestly 2 sessions a week aren't what are really going to make a difference for DS. Instead, my ability to work with the speech therapist, reinforcing things and maintaining certain types of interaction are why DS is REALLY improving now. Having the help of our speech therapist has made a big difference for me, helping me to better help my DS.
Like I said, you know your DD and I do not judge any decisions you make about her. But I do hope you make those decisions based on good information and it sounds to me like you maybe have an incorrect impression of what speech therapy has to entail. Like I said above, I definitely understand your fear of wrong labels and bad therapy. It really is important to find the right people to help you and your little one. But, with the right help, things can get much better, very quickly. You could even just hire a speech therapist to help you. She never even really needs to work with your DD (though she would have to see her at least enough to get a sense of what kinds of things might help her). Instead you could tell the therapist from the get go that you want some expert advice how to work with DD yourself. Any good, play-based therapist will be able to help (just make sure up front that they use only play-based approaches). The other option is to look for Hanen training in your area. This is a class that teaches YOU how to be your child's therapist. That class uses the "It Takes Two To Talk" book I suggested (which I haven't found anywhere for less then 65$!! bleh) as its text book and it is really good at helping you come up with techniques that can be tailored to your DD's specific strengths and weaknesses. I took this course and it was paid for entirely by our Early Intervention system (after I got pretty pissed about the way DS was treated but I know many school districts and EI folks will pay for this kind of thing).
No matter what, I definitely understand how frustrating it can be sometimes to not be able to communicate well with your little one. As a total aside, and I'm definitely NOT trying to diagnose your DD since I have clearly never met her and have only very bare bones description of her language, but your description is really text book verbal apraxia. Laughing at language demands and the appearance of just not wanting to talk are actually 2 of the things our dev ped listed when she was trying to determine if DS has this issue (which he might, though it is not clear because of his receptive delays). These are called avoidance strategies. Another red flag for verbal apraxia is the ability to talk well in other modes - like singing. Because singing uses a different part of the brain, many apraxic kids who can't talk very well, can in fact have whole conversations by singing what they want to say. Many parents of children with verbal apraxia believe that their children are just stubborn and "won't talk," especially because they are clearly smart and not delayed in any other way. I'm not saying that is what is up with your DD, but I do want you to know that the things you describe are very suggestive.
But no matter what is going on with your little one, I can't imagine a little therapy with the right person hurting and, who knows, it might help? No matter what choices you make, I support your right to make them without any judgement from me and wish you and your DD the best.