The purpose of an engagement is to establish a confirmatory level of commitment to allow publicity of the relationship for community approval, as well as planning towards the wedding ceremony where the marriage covenant is established.
Since the purpose is to increase commitment, it should be pursued in a way that does that.
That means it shouldn’t be a true “surprise” because that means you aren’t necessarily on the same page about your relationship status or progress. And it shouldn’t done publicly (at least for the true proposal vs a public one afterwards), which adds social pressure etc.
Ideally pre-engagement counseling has been completed successfully, and then there’s a long open ended meeting to review and recap the dating process so far, discussion of any unresolved issues.
At the end of that meeting, there’s a sober question of the man on behalf of the woman which can be worded different ways, but is generally asking:
“Will you bear this burden with me for as long as we live?”
Then after that point, they are agreeing to be married and can move forward with more confidence towards marriage.
Ideally that stage should function about the same as saying you’re married without the functional reality. And biblically and historically, the “betrothal” stage would be closer to marriage than an engagement.
The engagement phase should only last as long as it takes to successfully enter marriage (which should be defined before entering). Usually a period of a couple months or more.
The largest outcome of the engagement process is that marriage is a seamless as possible transition from the engagement phase practically speaking.