Can I let my streptococcus infection just run its course instead of taking antibiotics?

Bad idea. You do not state what type of streptococcal infection you have. Some streptococcal infections can become life threatening without treatment. Group a strep. Infections can cause rheumatic fever if not treated appropriately.
Antibiotics -yes. For most sore throats, you're right, antibiotics aren't needed. Strep is different. While the sore throat might go away on it's own, you're at higher risk for kidney and heart problems down the road if it's not treated with penicillin. Please, if you've been diagnosed with strep, get treated!