There are those who say that the Mawlid celebration, in the way it is seen today, is Bid'ah (innovation) because it was not present in the time of the Prophet and the Sahabah (Companions). It is clear that these people have not understood what Bid'ah is.

The word Bid'ah is used both in its linguistic sense and its technical sense.
Something that was not present in the Prophet's time can be called Bid'ah in the linguistic sense. However, can it be called Bid'ah in the technical sense? Anyone who knows the basics of Islam will understand that it cannot be called so.

Hafiz Ibn Hajar al-Asqalani (﵀) says:
'Technically, Bid'ah refers to matters newly introduced without any basis in Shari'ah (Islamic Law). Matters that conform to the fundamental principles of Shari'ah cannot technically be called Bid'ah."

والمحدثات بِفَتْحِ الدَّالِّ جَمْعُ مُحْدَثَةٍ وَالْمُرَادُ بِهَا مَا أُحْدِثَ وَلَيْسَ لَهُ أَصْلٌ فِي الشَّرْعِ وَيُسَمَّى فِي عُرْفِ الشَّرْعِ بِدْعَةً وَمَا كَانَ لَهُ أَصْلٌ يَدُلُّ عَلَيْهِ الشَّرْعُ فَلَيْسَ بِبِدْعَةٍ فَالْبِدْعَةُ فِي عُرْفِ الشَّرْعِ مَذْمُومَةٌ بِخِلَافِ اللُّغَةِ فَإِنَّ كُلَّ شَيْءٍ أُحْدِثَ عَلَى غَيْرِ مِثَالٍ يُسَمَّى بِدْعَةً سَوَاءٌ كَانَ مَحْمُودًا أَوْ مَذْمُومًا (فتح الباري ١٣/٢٥٣)

The same idea can be seen as stated by Imam Shafi'i (﵀):
"Newly introduced matters are of two types. That which opposes the fundamental principles of Islam – the Quran, Sunnah, or Ijma' (consensus) – this is the misguided Bid'ah. Good things newly introduced in a way that does not oppose these fundamental principles will not be included in the category of blameworthy Bid'ah."

وَجَاءَ عَنِ الشَّافِعِيِّ أَيْضًا مَا أَخْرَجَهُ الْبَيْهَقِيُّ فِي مَنَاقِبِهِ قَالَ الْمُحْدَثَاتُ ضَرْبَانِ مَا أُحْدِثُ يُخَالِفُ كِتَابًا أَوْ سُنَّةً أَوْ أَثَرًا أَوْ إِجْمَاعًا فَهَذِهِ بِدْعَةُ الضَّلَالِ وَمَا أُحْدِثُ مِنَ الْخَيْرِ لَا يُخَالِفُ شَيْئًا مِنْ ذَلِكَ فَهَذِهِ مُحْدَثَةٌ غَيْرُ مَذْمُومَةٍ انْتَهَى (فتح الباري ١٣/٢٥٣) 

Now, consider:
The Mawlid, celebrated with joy on the birthday of the blessed Prophet (ﷺ), which principles of Islam does it oppose? The reality is that even those who argue that it is Kufr (disbelief) and Bid'ah do not claim that such a celebration opposes any fundamental principles of Islam. Rather, what they themselves argue is that the Prophet or the Sahabah did not conduct such a celebration. We have already clarified that the Imams have established that just because the Prophet or the Sahabah did not do something, it will not be included in misguided Bid'ah.
Those who engage in this act of branding all the Muslims of the world as innovators (Bid'atis) for the sake of loving the blessed Prophet should think. May Allah grant you guidance.