Experts tend not to use the word 風邪 in serious documents in the first place, because there is no strict definition which they can rely on. Even Wikipedia says "医学的な(風邪の)定義は文献によって異なっている". In clinical practice, a wide variety of diseases cause cold-like symptoms, so physicians tend to use a bit vaguer but safer terms such as かぜ症候群 (症候群 = syndrome), 急性上気道炎 or 普通感冒.
According to The Japanese Respiratory Society, most (but not all) かぜ症候群 cases are viral, of course, and thus antibiotics are not effective. But they say some of them are actually bacterial, although antibiotics are still generally unnecessary anyway (unless complicated with secondary infections and/or other diseases). Many Japanese articles (like this one) acknowledge the existence of 細菌性の風邪 ("bacterial 風邪"). On the other hand, many English sources (incl. Wikipedia, Mayo Clinic and WebMD) seem to say "common cold" is strictly viral, so Japanese 風邪 or かぜ症候群 may have a broader sense than "common cold".
That said, antibiotics are not necessary regardless of the type of microorganisms, so I agree with your attitude towards 風邪 :-) You can find tons of Japanese articles saying how improperly some Japanese people demand the prescription of antibiotics once they feel slightly sick.