Can you talk to any of his other partners about this? As you say you've been avoiding some of the harder conversations when the topic is "what are we, what can we be" Perhaps he is avoiding directly asking his other partners if they feel like adding another relationship will be problematic for them, and discussing what it would mean.
My boyfriend has a wife, and another long term partner. When our relationship ended up being something that looked like it would be lasting more than a few months, his wife and I had a bit of discussion how two additional "serious" relationships for him was probably pushing it time wise. I "think" it is being managed OK and everybody is tolerating/getting their needs met for the most part. I don't like to think how everybody would be feeling if he wasn't staying conscious of how he was splitting his time.
If he started dating a fourth person, I can only guess what that would mean for his other two partners, but even if it ended up meaning our one date a week ended up being skipped once a month, or even cut short by a couple hours here and there so he could spend time with them, I am guessing that might be the end of our relationship. Because in our case we don't talk between dates, I don't think I'd feel like we were having the time to connect. I think I'd rather not see him at all than be feeling like I wasn't quite getting enough on an ongoing basis.
That said, he might think it'd be fine (or being avoiding really thinking about it practically), some or all of his partners might think it'd be fine in theory, but unless he knows what time requirements he'd have to commit to so you'd be happy dating him, and THEY know how he will have to alter his schedule with them to accommodate him dating you, it is just a guess. And just maybe one ore more of them would be glad to cut down the amount of time with him so he can see you and they can have more free time to do the laundry!
Happiness will never come to those who fail to appreciate what they already have.