Yes, it is necessary to inject the control group, as you may otherwise be looking at non-specific effects of inflammation caused by an injection of any substance. Prolotherapy, for example, works by injecting an irritant that causes inflammation which results in formation of collagen (scarring) thus strengthening weakened tendons and ligaments.
Another very important issue, of course, is the placebo effect. A person not receiving an injection would know that they are not being treated and thus not expect to get better.
I can’t comment on differences in protocol and skill levels between different practitioners, but it is possible that the anecdotal evidence of effectiveness Charlie and you describe may have been due to such a placebo effect.
Alternatively, the improvement observed may also have been due to other treatment modalities used concurrently. All these factors need to be controlled for in a good study, which seems to have been the case with this one.
Having looked at the actual study, it seems to have been quite well designed. The patients were suffering from chronic tendinopathy (so may have had symptoms for longer than the athletes you and Charlie are referring to). The last assessment took place 24 weeks after intervention, so long-term follow-up was done. The treated patients did report slightly more improvement (score of 21.7 vs 20.5), but this difference was not significant. The injections were conducted by a sports medicine physician and “the PRP preparation and injection was performed as the usual generally accepted procedure in daily clinical practice.”