There is no legislation or official guideline to cover the "stuffed" Mass (Proper and Hymn together in a single "slot.") So the best you have to go on is opinion and practicalities.

Is the Offertory hymn covering (primarily) the collection? Is then the soft organ playing covering the "Preparation of the Gifts" (or whatever it's called)? Do people really sing the Offertory hymn, or are they mostly busy writing checks and fidgeting with envelopes?

IT seems to me that in cases where both the Proper and a hymn are used together, their order should be determined by what action they may be accompanying. For example: If there is a clear activity distinction between collecting money and preparing the altar, then I would likely do the hymn for the first action, and the proper during the second. Similarly, singing a Processional Hymn first, followed by the Proper Introit during the clergy's entrance into the sanctuary makes more sense.
On the other hand (for example, at communion) when both the Proper and the Hymn are accompanying the same action (the reception of Communion by various people) it seems you'd want to begin with the Proper, to make it clear that it is the normative choice and that the hymn has been added. (Similar to the addition of a motet after the Proper in more traditionalist praxis).

On the other hand, if no one sings the Offertory in the first place, I would replace it with only the Proper, in settings ranging a variety of choral styles (chant, polyphony, Anglican chant, etc, etc).

I would never take singing away from a congregation that does it well and with enthusiasm. But I would not continue to push a congregational-song model on a community that seems uninterested in it.

But as I said- this is all largely a matter of opinion.

I always pick an offertory hymn that will be long enough to cover the entire (or most of the) action. I then pray ( aka improvise) until the priest is ready. If I was going to do the proper , I'd end the hymn as soon as possible to the gifts landing on the altar, and do it. They're all pretty short.

I second Adam's recommendation. This is what we do. It works very well.

The introit covers the time of the incensing of the altar, and the same occurs at the offertory. I've also noticed that the Offertory proper and the Communion proper "bookend" the Liturgy of the Eucharist quite nicely, where if you sing the Hymn second (which happens at other Masses), it doesn't seem to flow as well (at least not in my opinion).