To quote Smillie's1 opinion of the cause of the Osgood-Schlatter condition, "no acceptable evidence has ever been produced that the condition is a disease. It is a traction epiphysitis, the result of a single injury or repeated minor trauma."
In my sports medicine practice, I see several cases each year of complete detachment of the tibial tubercle epiphysis, and it is a well-recognized entity in sports medicine.
I know of no scientific rationale for the use of hydrocortisone or local anesthetics to promote the healing of Osgood-Schlatter condition or any other avulsion. To the contrary, the patient's awareness of his symptom level, not masked by local anesthetics or anti-inflammatory injections, is important in controlling activities, thus preventing avulsion of the epiphysis.
As long as syringes exist and bumps develop on young athletes' tibiae, there will be physicians who continue the practice of injection because they think Osgood-Schlatter
Cahill BR. Treatment of Osgood-Schlatter Injury-Reply. JAMA. 1978;240(3):212–213. doi:10.1001/jama.1978.03290030030009