What are you on about? Look at the top ten. All democracies.
1 Denmark
2 Norway
3 Switzerland
4 Netherlands
5 Sweden
6 Canada
7 Finland
8 Austria
9 Iceland
10 Australia
...
13 New Zealand
I think you would also recognize that the political system of democracy isn't the thing which is a determinant of happiness but rather the successes, fruits, and environment that can be delivered by it.
In that sense, I think conflating democracy with happiness is like conflating democracy with having a successful GDP -- i.e.: it's misconstrued at best and patently false at worst.
Unfortunately, it's impossible to do a t test of two countries that are identical in every regard apart from their political system. No such situation exists.
I do agree that there are certain aspects of democracy which can be incorporated into an autocratic system, but I do not think we can simply say that formal multiparty democracy is the simple cause for why the countries listed are the most happy.
It's like saying they're also the most happy because they are predominantly white, which is obviously untrue given there are many countries with caucasian populations which aren't particularly happy. Funnily enough, there are also many countries which are multiparty democracies which are also not that happy.
I'd say the ends to a happy population, are: potential for upwards mobility, demographics, safe/livable environment, work/life balance, social equality, infrastructure, among others.
I think I would be skeptical to say that democracy is the only means of providing that. For some countries democracy might be the best way of providing those ends, but for others it may not.