I use MCCC, and I have fiddled with settings to come up with what works for me to prevent the problem you describe.
I have abductions, and therefore male pregnancy disabled. I'm pretty sure that's in the "occults" module. This might not be something you want, I don't know.
I have only one age set to allow pregnancy, and that's YA. Thereby hopefully allowing most sims to at least see their grandchildren, if not great-grands.
I have homeless sims never allowed to move into a house, marry, or become pregnant. Only townies are allowed those privileges!
I have allowed children set to 2, so no townie will have more than 2 kids. Your actives can have as many as they wish, of course.
This should cut down on the population explosion, but of course you can always move in new sims, either of your own creation or from the gallery, for "fresh blood". You'll just have to make sure to leave a few houses open for them to move into. It might sometimes be necessary to maybe help a few elders shuffle off the mortal coil a little early, or maybe arrange a pool accident for particularly ugly or obnoxious sims. :D
I hope this helped, but your playstyle may not match mine at all. This is how I play, but the variations are pretty much limitless.
MCCC will seem intimidating at first. Start a new save and play around, get used to the settings. Also, Deaderpool's discord is always active and the folks there are beyond nice and helpful. Don't be scared to ask for help.