According to history available it is not proved prophet celebrated Mawalid.
But this does not mean Mawalid is Bidat.
Prophet did not use cell phone and computer. so it is Haram? no but it is Mubah because we do not have any evidence prophet made it Haram.
While we do not have an evidence prophet made something Haram we are not allowed to make it Haram only because we do not have any evidence to show prophet made it Halal.
Also maybe prophet did something but its hadith did not survive in 1400 years in history and wars and for different reasons. currently I have no evidence they celebrated mawalid. also we have no evidence they banned it. so banning what prophet did not ban is Bidat. when there is no evidence for ban or permit of an action in Islam it is considered Mubah. not Haram. changing something from Mubah to Haram is Bidat.
According to Shia Islam, it no Bidat. (ref)
According to sunni scholars, there is a difference of opinion. Some, like Al-Azhar University of Egypt consider it no Bidat. (Arabic ref, English ref) Others, like Ibn Taymiyah and Muhammad Inn Abdul-Wahhab), consider it Bidat. (ref)
Mawalid of family members is not Islamic but still is not Harm while it does not contain a Haram act. its like using computer. not banned and not recommended in Islam. but some Shia Hadith from Imams for shia like Imam Sadiq a.s says: our followers are happy with our happiness and are sad with our sadness. can you prove why Mawalid is Bidat?
According to Shia Islam when there is no evidence for one action to be Halal or Haram it is Mubah and not considered ad Bidat because there is no positive or negative rule for ir in Islam.
Cultures change over time and prophet said leave cultures not conflicting Islam. if prophet did not celebrate Mawalid it was not usual at that time.
Milawd itself is not done while claimed to be part of Islam. but it is used as a tool for advertising and promoting Islam and teachings of prophet. when you give free foods to people at Mawalid it attracts other people to Islam and make interest in them to learn about prophet and Islam
Definition of Bidah [Rasool Allah (saww) vs. Salafies]