You're right, but that wasn't a problem exclusive to Microsoft. Most ps2 exclusives were also 3rd parties, and when those 3rd parties preferred the 360 over the ps3, sony had to do what microsoft is doing now and build their internal studios up.Microsoft never had lots of studios to work on games until recently. Most of the “exclsuives” people remember from 360 were done by 3rd parties and happened out of circumstance I.E. xbox having a year headstart and PS3 being a bytch to develop for.
As the industry moved away from 3rd party exclusives microsft didn’t have anyone to make their games anymore, so they went through a drought until xbox leadership could convince microsft to buy/build some damn studios.
Microsoft should've done it a decade ago, but like you said, Xbox leadership had to pull teeth to get microsoft to do what should've been done with the launch of the X1