I like the idea: you get to engage with your listeners without prioritising their questions/feedback based on when it was sent, potentially meaning that any email can be eventually read. This is especially powerful given that the podcast and hosts, while indeed discussing news and goings-on, are ready and willing to tackle deeper issues and questions regarding the medium, so many emails go for those types of questions rather than just about what happened that particular week.
However, since the podcasts are short, and only a handful of emails get answered every time, what this means is now there is a huge backlog (I am assuming). And then the hosts pick random numbers, which are presumably in chronological order? It is a bit clunky. So I think that while the idea and the thinking behind it is good, the execution could be better.
I don’t pretend to know how to “solve” this, but yes, a monthly clean up seems like a good option, especially since I doubt that you will want to change the format and length of the podcast - it is well established now and flows naturally. An extra podcast that is geared at just answering questions could be a bit too much in terms of extra preparation and work perhaps, and will almost definitely increase the amount of emails sent in.