Without question, your recommendation is cleaner, less busy, more subtle. The choices of action are fewer, and therefore more clear.
Do they really want that big head panel to appear in addition to your proposed menu and time bar? If so, and I understand what elements are links, then it seems to me, too, that there's needless and confusing redundancy. If they want their big head panel instead of your cleaner approach, that would eliminate the duplication. But even then, I'd move their big bar down to the bottom because the photo of the island is the pull, not their text (man, that is a big bar!). And move the "4€" into the "play now" box.
In sum, as long as they want to replace rather than duplicate, I'd say there's not a lot to choose between them from a psychological standpoint. Theirs, especially at the size shown, is a "bullhorn" approach and will appeal to people who are comfortable with tv and tabloid news because those two modalities also use a bullhorn approach. So if that's their target demographic, go with it. But if they want to target a more upscale demographic, they should avoid the bullhorn and go for subtle. Or you could split the difference by shrinking their bar to a less-intrusive size, moving it to the bottom, integrating it with your design elements, etc.