I'm not being snarky but I genuinely don't understand situations like this. If you want to get married, wouldn't you say (man or woman) "shall we get married?" And why does it have to be the man?! same sex relationships must really mess with people's minds.  What do you think lesbians do?
"Do you want to get married?" The answer is either yes or no. If it's no, then you have the conversation about why not, eg someone wants to wait as they want to save to have a honeymoon at xxx place, or they don't feel ready to commit to marriage just yet - whatever reason they refuse marriage, then there's a wider discussion about compatibility and future of relationship.
If it's yes then surely you are just pleased and you book it.
I also don't get the whole "shall we get engaged" as a stand alone thing, or "I want to be engaged but not married yet" thing. I'm trying not to judge as I do understand people like different things  but I genuinely can't get my head round that. If you agree to get married then the period between that point and your ceremony is an engagement period, I get that. But Ivf had situations where someone shows me an engagement ring and I have said "lovely ring, when are you getting married?" and they've said oh we don't know yet we may just be engaged for a couple of years, then decide". It's like the engagement is an end in itself, not an automaticside process of getting married (if that makes sense?)
I just said to DH shall we get married, he said yes and (I don't know how it's done these days) we went to a registry office and the earliest date was six months later so we booked it and got married). Thirty years later we are still married. It seems much more complicated now. Admittedly I am bitter as I once bought some really nice crystal glasses for an engagement party present, and they didn't get married. I don't know who kept the glasses!
Don't get me wrong, I love celebrating a wedding, but to me it's quite straightforward - you want to be married or you don't, there isn't some special gateway to being married that is unlocked by a magical process that can only be initiated by men!
"Shall we get married?" I would start there. You can still say it even if you don't have a penis...