Relevant Hadit Sahih on the subject of bid'ah are
The Messenger of Allah said: 'Whoever initiates a good practice that
is followed, he will receive a perfect reward for that, and a reward
equivalent to that of those who follow it, without that detracting
from their reward in the slightest. And whoever introduces a bad
practice that is followed, he will receive the complete burden of sin
for that, and a burden of sin equivalent to that of those who follow
it without that detracting from their burden in the slightest.
He who innovates things in our affairs for which there is no valid
(reason) and these are to be rejected.
We can draw thereof the conclusion that the Prophet ﷺ did not reject innovation in general but depending on whether it is a good innovation or a bad one. There is also a region where an innovation is not entirely bad but still not esteemed a good Sunnah, for example as explained in this fatwa, rejecting a common prayer after the regular salat. However, it does not reject du'a after salat but a prayer spoken loudly and in common as a supplement to the common prayer.
There is the aspect of innovation to Islam, as it is added as an obligatory part to the common prayer in the mosque.
And it is qualified not a good reason because Du'a is established as an individual prayer. So this is not real du'a.
Approaching your question:
In the time of the Prophet ﷺ there was hardly any copy of the Qur'an available and most people were not able to read (for this reason, the Qur'an had to be learned and recited as we do it until present). A tradition to gather and read the Qur'an could not have been established. In this sense, reading the Quran in common is an innovation. However, the Qur'an was meant to become a book and a book is meant to be read. Hence, this is certainly a good innovation conform to the intention of the message of the Qur'an.
In this sense, gathering to read the Qur'an is innovation (bid'ah), but in the sense of good Sunnah. I see no reason why a good Sunnah should become a bad one if it becomes a tradition.
P.S. People may wonder whether I did not cite Sunan an-Nasai 1578. It is because this text, although it sounds absolute, has a certain scope. From the context, it refers to the Book of Allah and the guidance of Muhammad ﷺ. It is possible that it actually refers to inventing Qur'an verses and false Hadit which is the minimum scope to apply and it is evident that such action is major sin. Its scope of this text is often understood in a wider sense. Scolars have differing opinions on this topic. On the topic asked, I would rather give priority to IbnMajah 1,204, as said before.