There is no way an invitation can contain all the information guests might want to have. Sure officially it has, or gives clues to, all the info a guest should have, but with people being so scattered, and reading the nuances of invitations isn't as easy as it might have been in days past. Things like dress code - traditionally the formality of the invite would tell the guest how to dress. But these days, that's not always a clue. Also traditionally one would ask the friends, family or wedding party of the HC about gifts and registries, but these days its very possible many guests don't know or have contact info for any of the friends, family or wedding party.
I think many modern engaged couples do wedding websites. I can't imagine how the website is "putting their wedding out there for people who weren't invited" - I've never known anyone to publicize the web address to anyone except invited guests. Its a great way to get random info out to guests. My wedding website let people know my ceremony would be outside on a rooftop and asked they plan for windy weather (it was warm, but windy). I also listed the hotel we'd booked a block of discounted rooms at, and that we would provide free transportation to/from the hotel and wedding. I gave people some suggestions about fun things to do in the area, and I mentioned the history of the building the wedding was in. There was also a photo of the outside of my location, which was handy for people driving - even with directions its always useful to know what you looking for.
Either way its always been, and IMO always should be, considered bad etiquette to mention what one wants on or in one's invitation. I am of the personal opinion "no gifts" or some variation is ok but not any mention of getting gifts (no asking for donations, cash in lieu of stuff, mentioning registries, etc). The only exception is for showers where the guest of honor is not the one issuing the invites, and the sole purpose of the party is to shower the GOH with gifts.