It has been a while, but Xbox was THE gaming brand back in PS360 generation. In the late 2000's and early 2010's, when you thought of Call of Duty, Madden, and NBA 2K, you thought that they were Xbox 360 games. It didn't matter whether those games were on the PS3 or not, tv commercials ended with the 360 logo. Grand Theft Auto IV launched at a 2:1 ratio in favor of Xbox 360.
Heck, most of EVO was played on 360's during that time.
Two crowds that Sony went out of their way to court at the beginning of the PS4 days were the COD and e-sports crowds and that continues to pay dividends to this day.
And that doesn't even factor in how much stronger the Xbox exclusives were like you said, especially during the early 360 days. That was the time of peak Halo. Forza was giving Gran Turismo a serious run for its money. Fable and Gears were hot items as well. Things started to shift towards the end of the generation as Sony's first party offerings started to become more popular.
The Xbox brand continues to suffer for the sins committed during the early Xbox One days.