That's exactly what they're doing. Microsoft is leveraging their advantage over Sony (they have a lot more money to throw around, because outside the PS4, Sony is not doing great financially), to try and combat Sony's advantage over them (tons of first party content), in addition to Sony's PR advantage.
As far as PR goes, Microsoft is doing everything they can. They've basically axed everything consumers didn't like. But it takes time to rebuild trust and brand value. One of the best ways to do it in the gaming industry, is offer more great games than the other guys, that you can only get there. Do it consistently enough, for long enough, feelings start to change. Take a look at Sony at the PS3 launch, to Sony when Last of Us released.
Now, the argument is there that Microsoft is taking a shortcut while Sony invested for the long haul. That's true. Hopefully Microsoft will start to reinvest in their own internal software. The people that axed most of their first party stuff and set the initial vision of the Xbox One are pretty much gone now. Phil Spencer, who a lot of people in the industry seem to have a good opinion of, seems to be trying to overhaul this whole clusterfuck. I hope that means we see a return to the kind of thing we saw in the initial J Allard days, when they were making deals with everybody from Oddworld to Double Fine to make new games for them.
Until then, they have to buy time but give consumers a reason to want an Xbox One over PS4. They have the money to spend on exclusives and timed exclusives, so why wouldn't a company try to leverage that as much as they can to increase the perceived value of their product?