Yes, it's perfectly possible (using something called a CSS media query) to serve different themes to different devices. Desktop-like devices (including laptops) - serve the old one, or similar, with vertical scrolling. Mobile devices - use the current swipey version.

I look after a website for a wolf centre (as a voluntary thing, no money changes hands), and funnily enough the owner wanted what we see on Aerolopa - a massive full screen image that you have to scroll past to get to the actual content. I managed to persuade her that no, that isn't a good idea - the basics of user interface design are that you should be able to get to where you want to go to quickly and efficiently, with the minimum of scrolling past pretty pictures and with direct links to important content right there on the homepage. You can see quite clearly by using analytics that the majority of users - whether mobile or desktop - just won't scroll much, they'd rather click links in front of them rather than play "hunt the content amongst the artwork".

I used Bootstrap to handle the layout, incidentally, so you get a trendy "hamburger" menu on mobile, but on desktop you get a traditional menu bar, with dropdowns for page selection.

One size really doesn't fit all IMO, and just increases frustration. Desktop users really don't want to faff around trying to find what they want, especially with wretched horizontal scrolling - on a screen which is likely to be wider than it is tall - and mobile users don't want to squint at tiny writing and impossible-to-tap miniature buttons.