Sorry, what do you mean with "they also clearly want to compete with the traditional desktop applications"?
Both in OS and in productivity they still dominate that space, despite declining market share... if by compete you mean they want to protect their position then I agree, and the convergence of mobile with traditional is the biggest threat they face, hence their strategy.
I am curious what you base the statement that "MS has a greater interest in the windows gaming scene this time around". As you yourself say, gaming is popular in app stores, hence IMO any apparent gaming focus is a consequence of their need for a store, not the other way around. As well you actually take it one step further by considering an interest to compete in game retailing with Steam / GOG (albeit digital retailing) not on gaming per se.
Let's put it this way, until MS reverses direction on closure of dedicated gaming studios (even AMZ now has gaming studios) I see no indication of a specific priority towards gaming on their part. The creation of the XBox (associated with GFWL) was the only time MS actually focused on gaming as a priority - IMO they moved there because they felt consoles (Sony, Nintendo, etc...) were threatening to fragment their market and they reacted to that.
It was a strategy with mixed results, since their actual main threats emerged from different OSs so to speak, from Apple and Google (Google is moving in the direction of a virtualized OS, where the desktop is cloud based). One can look at the fact MS still has the upper hand over Linux and iOS in gaming as a sign the strategy paid off to some extent, but on the other hand the whole gaming involvement was a distraction and I think responsible for the loss of focus that permitted Apple and Linux to gain ground against MS in the past 15 years.
If anything MS is actually likely divesting of specific focus on gaming and replacing it with specific focus on hardware. So a better integrated store front isn't enough evidence of a renewed MS focus on gaming, and neither the fact their most recent Surface is powerful enough that it could be a nice gaming machine, and neither a renewed push to engage app developers to consider Win rather than just iOS and Android.
I think all of those facts are more easily explained by their need to protect their traditional market spaces by getting some position - any position - in mobility. They are retrenching and doubling down on convergence basically - I don't see gaming coming into that decision at all - it's just impacted by it. Whether this strategy will be good for them or not, who knows...
Here, have some analysis from when Win10 launched,
maybe you'll find it interesting.