Well...actually...if you just get assessed in the US, you will be bound to their culture... it's not like the US is beyond bias either...and they won't be able to tell you how to access the local therapy here in Switzerland that your child may benefit from.
You do have several other options, which is to go via your Swiss Paediatrician for a referral, find someone privately in Switzerland for yourself, use one of the many child/family psychology clinics available throughout the country. In fact, our Paediatrician was disappointed when she found out that the school psychology had been involved and she didn't know about it, and that we were self-referring and choosing ourselves...she partly had a different opinion on the course of action, and partly told us that she could have saved us hassle by using her own contact network to refer more efficiently...
The big advantage of using the school psychologist is that if they recommend/refer then it will be at no or low cost to you. The school psychology in our town has paid for thousands of francs of therapy and assessment for our children, which there is no way we could afford ourselves...the funding does 'run out' after about 2 years with the local town, then you have to go to the IV (disability insurance) if it's going to be a long-term problem that needs ongoing support, or sporadic support when needed which can be paid for by the health insurance, which only covers a proportion of therapy and only if prescribed by a Swiss psychiatrist.
Sure, if you feel more comfortable using a private psychologist or psychiatrist in your home country to do the initial testing, this can be reassuring (and take it out of the school system) but if you need ongoing help then it's going to be very expensive unless it's done via the school system or health insurance.
As for the fear of labelling, I have no idea, for my kids, they go the labels they needed, and the 'Nachteilausgleich' (special consideration) that my daughter got helped keep her in the highest academic level, where intellectually she belongs even though her exam performance is poor (the key issue for her is exam performance and social confidence / classroom confidence)...
The school psychologist actually doesn't do a lot - they are first point of referral and can meet with the parents and the teacher to discuss the challenges the child is having, and then make recommendations for further testing or referral. They can't even 'prescribe' therapy. If, for example, they think that psycho-motorik therapy could be helpful (PsM will look at fine motor/balance/gross-motor and movement, and can do movement therapy as well as art/dance/music type therapy, and possibly have a speech therapist within their team)... but in fact it was the PsM therapist who did the assessment and then made the recommendation for therapy which went back to the school psychologist for approval. Same with 1:1 therapy, they sourced the therapist who then did the assessment but it was up to that therapist to write summary reports every three months to the school psychology who then approved the payment for the therapy...
Hope that makes sense.
With hind-sight I'd much rather have my children labelled with labels like 'dyslexia, speech delay, gifted, anxiety,depression, autism spectrum, ADHD' than 'difficult, oppositional, violent, underperforming, bad parenting'....
And by law, the school can't refer without your permission. Same with the other medical stuff (the kids bring forms home for things like hearing tests).
The other reason why the teacher will ask first, is because you may have already done some testing or have things going on in the background... we are not obligated to tell the teacher or the school everything that is going on for us and our kids...in fact, we were advised not to give the full reports from the Swiss KJPD, as they are highly technical and also may not be relevant 6 months later, and instead to try to communicate the key challenges our kids have and what strategies have worked in the past, and what we want from the teacher in terms of our own expectations as parents.
The teacher can't work on everything (nor can the parent) with our kids teachers we focus on some things and agree to not worry about other things, and measure progress together over time.