Definitely bypass the engagement. A lot of people get hung up on the bended knee hearts and flowers stuff but it doesn’t change anything.
You want to get married. You’ve been clear about this. He’s in agreement in principle. Tell him you want to plan your wedding and just get on with it. That’s not meant to sound snappy but being married is something that you want and he either does too, in which case do it, or he doesn’t in which case have a serious think about whether you want to stay with someone whose plans for the future don’t line up with your own.
Now is the time. “I’m so excited about buying our first home together. We spoke about getting married and I said I was keen to do so before we bought the house. Let’s pick a date”.
You wouldn’t want to marry someone who didn’t want to marry you. But from what he’s said you believe he does. Don’t get blinded by him having to ask you. You’re adults. It’s a big bloody deal. Being married is so much more than how someone asked someone else or what sort of dress you wear, what cake you have. You wouldn’t leave any other big life changing decisions up to him would you? You wouldn’t have said let’s buy a house one day, look at some possibilities in an estate agents window together then leave it up to him to decide what, where and when and present you with a set of keys and a signed mortgage agreement 2 years later.
It’s your life too. You know what you want. Go get it. I hope he’s on the same page as you and it’s a productive conversation 
So often women feel embarrassed or awkward about discussing marriage. It’s not something you earn if you’re good enough, or that one day he’ll bestow on you when he’s ready. You’re buying a house. I imagine you’ve had many discussions, put a lot of thought into. It’s a massive financial commitment! You need to be sure you’re doing it with someone who’s committed to you as you are them, and for you commitment means marriage.
There’s another post on here from someone who gave her partner an ultimatum, he proposed a year ago and hasn’t discussed actually getting married since.
Having a pretty ring on your finger is great but it’s not being married. It’s an intention that doesn’t mean anything till you’ve picked a date and are committed to being married.