It's hard to imagine a world where it would not, since the effects of that one change are so wide-spread and pervasive, but I'll try my best.
For one, the original Xbox might not have been made, and if it was, it probably would have failed. XBL was a big part of their strategy for consoles (and still is), and helped drive the success of latter Halo games, which helped sell the early platform. Without XBL, the Xbox as we know it would not exist.
In addition, DLC and patches would not exist on consoles. Games would either ship hopelessly broken or with inflated debugging periods. Games would come less frequently, and what can be fixed simply today with a day-one patch would be catastrpohic.
Without DLC and balance-patches, local multiplayer would be the only way to play, and without a way to add content, would be far less long-lived. Without the social-aspect of online for consoles today, the multiplayer scene would be much smaller, and eSports as we know it would not emerge on consoles until much later, if it ever did.
To compensate, more on-disk locked content so publishers could get the effect of DLC, but without the benefits it provides. The same sort of online seen on consoles would evolve on PC, albeit smaller. PC would eventually kill consoles, as the lack of online would be too stifling for publishers compared to the advantages offered on that platform. Sony would eventually shutter its EE division, though Nintendo would press on, with their handheld systems keeping their sales afloat.
Gaming quality would not appreciably improve, since the realities of consoles that produced much of the oft-spoken complaints of the industry are just as much, if not more true, of PC. Gaming would be much more exclusive due to the realities of PC gaming, and digital distribution would have taken hold much faster, with predictable results for Gamestop. Microsoft would be in a dominating position, due to them being the only significant platform holder.