The restaurant version seems a little weird to me, and likely to impose on the restaurant--disorganized, people not calling ahead, everyone sitting in the same section, etc..
Otherwise I think it depends on what Adam and Eve really want. If they definitely want a party, they need to host the party they can afford. That could be a light cake-and-punch reception in a public building or a friend's home; or it could be a potluck where everyone (including them) contribute something, if that's generally done in their social circle. Or, honestly, sometimes there are people (like parents, perhaps) that you can say to, "We would really like something bigger, but we can't afford it. Would you be able to help us out, since you also want us to have something bigger?" I don't think this is strictly polite and it should not be broadcast on Facebook to all and sundry, but there may be some people where the relationship permits that kind of conversation.
If they're cool with a gathering of people but would also be cool being alone, the "bring your own meal" picnic is kind of nice. Just in case I think they should have some extra soft drinks and maybe cups/napkins/plates/silverware on hand--that kind of stuff can be bought cheaply and doesn't go bad, so they can use it themselves if few people attend, and it shows they've given some thought to the people they mentioned the event to.
What about something like, "After the ceremony we'll be at Bar X from 5pm to 7pm, if anyone wants to drop by." No food involved, but people can stop in for ten minutes, have a beer, and wish them well; and it would be casual enough that I don't think people who show up would expect anything paid for or organized. That really depends on your group of guests though--Grandma or your cousins with young kids or whatever might not appreciate this type of "reception" much.
If they really don't want a party but are feeling family pressure, I think they need to stand up and say, "We're having the wedding that we want." The end, no further options given. I think if it's Great Aunt Matilda saying, "You need to have a 'proper' wedding reception!" she's not going to be satisfied with a bring-your-own-meal picnic or a potluck, and possibly not even cake-and-punch, you know? So unless she's putting up the money and the HC agree to do whatever she wants as long as she pays the bill, she's not going to be happy (and my guess is she also won't be happy to pay!). So Great Aunt Matilda is never going to be happy, basically, in which case the HC might as well do what makes them happy (with the caveat of showing consideration for their guests, if they invite guests).