Abs, love, I'm so sorry it's so hard.
This is very very familiar.
And I know it's hard but here is the honest truth. This man is a coward. Do you remember what I wrote about the instinct being to keep both things? That's what's happening. And it will go on happening. He is saying to her at the moment that he needs to make sure you are strong enough mentally for him to go, and that if it's your decision and you own it it will bode best for his future relationship with his children. Then he gets to leave and not feel like the cheating arse but like a romantic who fell in love and you couldn't deal with it and he had no choice but to go. It's all about preserving his sense of self and a narrative that makes sense to him being a good guy. Inside, he is thinking to himself that sure, he really loves OW, but maybe some miracle will happen and you will have a conversation where it all becomes clear.
If a cheater does not return to the relationship immediately with a renewed sense of purpose, commitment, apology and gratitude, all coming from them, then the forecast looks very bad indeed. The likely scenario is he will continue the affair in some form, whilst ostensibly 'working' on things with you, having good times and bad times on both sides until forced to choose by either you or OW, properly. If you do that now you will save yourself years of heartache.
It will only work if he is passionate about making it work and can't believe he nearly threw it all away and immediately cuts all contact with OW. That's just how it is. If that's not how it is, you have to answer one question, will you do it now or will you do it later? The only motivator is true loss. He will either come to his sense and beg to return, or breathe a sigh of relief and leave for good. Either way you need to detach, move on, and cut contact with him.