Public weddings are also about the guests. The guests serve a function: they witness the promises, and tacitly agree to help support them in fulfilling it. (I've even been to ceremonies that make that explicit: the guests are asked to reply to a vow with "We do".)

Not every couple wants that. Some elope, and others hold ceremonies only because they think relatives and friends demand it of them. The latter is unfortunate, because it imposes a burden on the new couple that they shouldn't have to shoulder.

But the couple should consider that community can be a crucial part of the success of their marriage, in ways that they may not recognize (especially if they're young). It can benefit them to have the community at the ceremony and feel invested in the marriage (for however they define "community").

Having taken that into account, they should by all means do whatever the hell they want, since it's their damn wedding. But if they do want that kind of witness and tacit promise of support, they do need to make compromises with everybody involved.