I think Microsoft best interest was to have the best sales possible for the title. It is probably the end of 3rd party temporary exclusive or exclusive deal on Xbox One.
No, Microsofts best interest was to attract people to their platform and in light of their deal with CD, it was to convey the message to all the Tomb Raider fans out there that the next game was to be on Xbox and that that's the platform they will want/need to get if they want to play it. It's effectively what Sony has been building up since the PSone days - attracting lots of games to your platform, if possibly exclusive, and make people associate those games with your platform. Ultimately, that's what Sony did so well and why they have been effectively the top dog in this market since their entry, even during the generation of PS3 that showed them launch an overpriced, year late and severely underperforming console. Because people still associated the games they want to play with the Sony platform - either because they were still exclusive on that platform (the 1st and 2nd party franchises) or thanks to them still getting the support from publishers/developers.
As a playstation owner, even if the Xbox was twice as powerful as the PS4, I would not care for a second because the games don't attract me in the slightest and those that do aren't exclusive. In fact, the games exclusive on Sonys platform appeal to me way more than to be bothered about graphical differences. It's why I ultimately got the PS3 over a 360 and why my mind was made up of getting a PS4 way before the specs were even revealed. In short, for Microsoft to attract someone like me, they'd have to tempt me with games first - games foremost exclusive to their console. They either do that with significantly better hardware, with games that are leaps and bounds better than what I can get on the PlayStation or by exclusivity contracts. In other words; if suddenly most Sony associated franchises somehow ended up on the Xbox - I'd have to seriously consider which platform I get. But this of course won't happen by mere fact that most games I like to play are Sony first party studios. The other games are multi platform and the rare games I do envy Xbox gamers over... well, I can live with not being able to play them.
Given my nostalgic memory of playing Tomb Raiders 15 years ago and the fact that I liked the new reboot quite a bit, the exclusivity deal on the next Tomb Raider game IMO was a well played move. If the game had turned out to be better or more in line with my nostalgic memory and who knows, Microsoft had secured more of those exclusives and the hardware wasn't that bit behind, perhaps the prospect of owning a Xbox console wouldn't be all that off for me. But in light of Microsofts dilemma, they certainly were on the right move. Doesn't change though that I expected better from SquareEnix when they signed that exclusivity deal.
This post probably belongs in another thread, but ultimately, Microsoft has to work hard on getting better exclusive games on their platform. It's the only distinguishing feature and feature that ultimately matters to us gamers. They either achieve that by signing exclusivity deals like the one they did with ROTR (which might result in gamers hating them even more) or they should get to work by making more and better games gamers want to play. Franchises like Uncharted that people buy consoles for. Or Gran Turismo. Having Forza and Halo are steps in the right direction, but they need more than that to sway a market in their favor. The more I think about it, the more I think the success of the 360 was simply exaggerated thanks to Sonys overpriced, and year late mess of a console. If they had launched side by side, even at the higher price, I think the PS3 and X360 probably would have been rather similar to the sales we see now in PS4 vs Xbox One. Close in the US, absolute annihilation everywhere else.