My first thought is that your DD clearly doesn't have the same view of her spots as you do, or she wouldn't have been to see her GP about them - ie for her, they are not "not really bad" and she wants an effective treatment for them and her viewpoint should be respected - she has to live with them and the effect her skin has on her confidence and mood, not you.
My second thought is that you shouldn't worry too much about the press coverage of over-prescription of antibiotics. Tetralysal is a recognised antibiotic for the treatment of acne (or "spots"), so it is not being prescribed totally inappropriately here and should be effective against the bacteria causing your DD's spots. The press coverage relates more to the prescription of antibiotics for inappropriate reasons - eg giving them out to the patient who comes in with what is probably a viral infection, or at most a very self-limiting infection, but who insists that they be given something to help and therefore come out of their GP's surgery with a prescription for antibiotics that they think will make their sore throat/cough/red eye go away more quickly, but actually won't, because they don't even have a serious, if any, bacterial infection at that point. And then, self same person not even finishing the course of antibiotics they've been given because they get better quickly, despite all the warnings that you must NEVER stop a course of antibiotics early (unless you suffer a serious adverse reaction, of course), because that is likely to increase the risk of bacteria becoming antibiotic resistant.
However, my third thought is that of course it is natural for you to be concerned about any treatment that your DD has prescribed for her and you are going to want to understand how it works, how long she is likely to be taking it and what the possible side effects are (and that she really has considered the possible risks as well as the obvious benefits). I would therefore want to read the leaflet that goes with the antibiotic very carefully myself so that I felt I was fully informed and would also want to comfort myself that my DD had also carefully read the leaflet and informed herself of the possible side effects and what to do if she is unluckly enough to suffer from any of them. But at the end of the day, if your DD has considered the risks and benefits and still wants to take the treatment, I think it really should be your DD's decision, without emotional pressure from you to change her mind. Tetracycline-related treatments have been used on mild acne for years - I would be far less worried about this than any of the more potent treatments for acne, which really can have nasty side-effects. And I speak as someone non-medical who also had mild acne on her face as a teenager and who never suffered any ill effects from a similar antibiotic, but got a LOT of benefit from it.