There's a lot to untangle here, but I'm going to recklessly jump into this stack-reach discussion.
First, Mr. IGH is correct when he insists that stack and reach are the best way to compare bikes. If you want a new bike that fits the same as an old one, the best way to do it is to match stack and reach. That said, you can't compare stack and reach independently. Two bikes with the same reach but very different stack are going to fit very differently.
At the end of the day bike geometry does boil down to geometry. If you project each bike onto a piece of graph paper, you can make the fit the same by lining up the three contact points to be the same. Your new calculations regarding how head tube angle introduces a change in reach is spot on. I'm not sure what the calculation you used in the spreadsheet earlier was, but that was incorrect. In general, the effective reach will come down about one unit for every three units you increase the handlebar height.
However, there are some other things to consider here. One is that the Vaya is designed to fit a much more upright riding position than the DCD. You can probably make them fit the same with the right combination of spacers and stem, but whether or not you want to is a different question. I said earlier that lining up the three contact points will result in the two bikes having the same fit, but it will have other effects. Most notably, as you change the position of the handlebars you are also changing your weight distribution relative to the wheels. A bike's geometry is selected, at least in theory, to achieve a certain desired distribution of rider weight relative to the wheels of the bike. As you raise the handlebars, you are effectively shifting your center of gravity up and back. This will have some effect on the handling of the bike. It's probably within the range of being OK, but it's a difference to consider.
The thing is while matching stack and reach will duplicate the effective fit of the bike, that isn't the only fit that will work for you. Your body bends in a lot of places and as a result there is a range of positions that will work. The thing they have in common is that they achieve a balance where your weight is basically equally distributed in front of and behind the bottom bracket while the saddle is at the right height to allow your leg mechanics to function optimally.
At this point, it gets complicated in a hurry. A few things can be sketched out roughly. As you lean over more, your hips need to come back to compensate. If you don't want to change how much you lean over, there is still a range of handlebar positions that will work as you rotate your arms and bend your elbows. The key thing here is that as you move the bars higher they will feel closer because your arms extend further as you lift them. There's a very enlightening article about this here:
https://www.rivbike.com/kb_results.asp?ID=33
What I'm trying to say here is that I think buying a Double Cross and essentially trying to turn it into a Vaya with shorter chain stays is a bad idea. If you want something sportier, try the Double Cross with a different fit. Maybe test ride a few bikes in the shop with similar stack and reach to get a better point of comparison than the Vaya offers.
On the other hand, if you like the riding position of the Vaya, keep the Vaya and just get better tires. Absolutely nothing makes more difference in ride quality than switching to really good tires. The Paselas that Mr IGH mentioned are a decent tire at reasonable price. If you are willing to spend more, consider the Continental Grand Prix 4 Season. They cost about twice what the Paselas do, but I think the ride is significantly better and in my experience they have better puncture protection (though I didn't use Paselas for long, so I may have just had bad luck). There's supposed to be a 700x32 version of the GP 4 Seasons available this summer. It's at the top of my wish list.