That will get you in the front gate at a reputable facility, but won’t be sufficient for most facilities to allow you to offer professional services like riding or training on their property.
If you plan to skirt the insurance issue by cultivating backyard type clientele who accept or don’t understand the risk, how will they find you and why would they pay for your services instead of finding someone else who will ride for free or even pay for time in the saddle?
When I had several horses I needed help exercising it was easy to find capable adult and teen riders who were delighted to ride for free, and probably would have paid for the privilege if that had been the arrangement. Is there really a market for professional exercise riding?
As for going pro as a trainer at age 16, I’ve known exactly 2 people over the course of multiple decades in decent eventing and dressage barns who really had an aptitude for training at that age (meaning, had developed the timing, feel, understanding of the aids to be very effective riders and capable of teaching horses new things intuitively). Both of them were humble enough to realize that they needed a lot more experience to build that talent into a pro skill set. Again, is there really a market for trainers with the level of experience of a16 year old?
OP, if what you’re interested in is a long term career in horses, you’re better off staying in your working student position and building up skill and reputation under the tutelage of a good trainer. If what you want is some summer spending money you’re better off finding a conventional teen job or equine gigs that don’t involve so much liability (e.g. braiding for shows).