Advantages to Wizards:
1) greater exposure. Which is better: A) 1 company with a huge budget doing all the advertising and product creation or B) 1 company with a huge budget doing all the major advertising and product creation while 100 other companies produce products for your game that you do not have the time, energy, interest or resources to produce and spreading the word about your game (even if indirectly) using social media, word of mouth and kickstarter? All those little companies are small individually but essentially work to promote your game in a huge way in aggregate.
This would be the economics effect cited in the discussion about the OGL all those years ago: if you build a network around a core product aou're selling, the network feeds back to you increasing the sales of your core product. This effect might have been working with the D&D Trademark License, which required the producers to print something like "D&D PHB is required to use this product". But the OGL, where you don't have to give any reference to the producers of the core product?
2) Larger pool of designers that are highly experienced with your game to choose from. Take Mike Mearls for example. When Wizards hired him, he was a top notch designer and thoroughly experienced with 3.5. Had there never been an OGL, Mearls probably would have written for someone on some other game system (we'll never know for sure, but ...). Maybe it would have been Wizards, maybe not. But the one thing that is certain is that they would not have had as large of a pool of designers to pick from. The larger pool means the top producers are generally higher quality.
This might be an advantage if your strategy is to gain market share due to (percieved) higher quality or you have to fight for quality personnel to further your own product development.
3) Higher sales ... indirectly. Here's the science of the OGL: You play a game, you get bored and move onto something else. But if you have a large pool of options for that game, you are more likely to stick with it that game. Its the basic "replayability" argument of many video games and board games. So if a DM gets tired of playing basic fantasy and wants to fantasy with guns. If there is no option for guns in a Wizards book, the group will probably goto an entirely different game. But if a D&D Compatible Publisher makes a supplement on guns, then the group can stick with D&D. But the mage doesn't use a gun, he is still using spells. So he buys the latest mage expansion for new spells. And that is a sale Wizards would not have had had the group switched to a different game. While Wizards had not sold as much to that one group had they produced a gun book themselves, they had higher sales with the mage supplement then a book on guns will.
Your example only works if the hypothecial 3rd party publisher designs his book on guns in fantasy as strictly additional elements, say, a gunmen class with gunmen feats and, of course, guns. But what if he changes the rules for ranged attack in order to make his gunmen more realistcal or cooler? Suddenly this material is at odds with WotC books. Isn't it much better for the 3pp to take the OGL and run? Make a complete game of it instead of just an expansion?
Sure, there's convenience for the players involved, as they can play the new game as a variation of the one they already know.
EDIT: The reason Wizards' did not (in this hypothetical scenario) produce a gun book themselves is basic business: opportunity cost. By taking one opportunity, you are turning down another. This means that if Wizards has the resources to produce 5 splat books/year, they can do martial classes, roguish classes, spellcasters, elves/halflings, dragonborn/tiefling books. Or they could do guns, space hamsters, androids, incarnum and an Ancient Canadian setting book. Which of those opportunities gets them more money. The first. This means they get higher sales because they could ignore those opportunities despite that 1% of gamers begging for guns in their ancient canadian themed setting. Without a sizable Compatible market, that 1% of gamers will go elsewhere. Do that often enough and you lose your market share. So either you A) produce to a 1% niche at the expense of the 99% that are not interested in that product or B) you slowly lose customers. Or you could go C) encourage a vibrant Compatible market that can cover those kind of niche markets that you never will that ultimately keep people in the game.
You should notice A and B are both lose-lose while C is a win-win.
Your analysis leaves out an important fact: the 1% loss to your customer base isn't all the wandering taking place. You'll have:
A) New players/customers gained (non-gamers)
B) New players/customers gained by winning them from other companies
C) Players/customers lost who don't game anymore
D) Players/customers lost due to them beind dissatisfied with your/3pp offerings.
Your 1% is oart of point D). I can easily imagine a lot of business strategies focusing on point A) alone. Present your customers a cohesive product world. Change this product world to keep in touch with changes in your target population. Focus on trademark image rather than on subjective quality. Sell your core products to new customers.
Note: For me as a player the OGL is cool. I relish the choice it gives me and I enjoy the sometimes quirky results it produces. Also, I firmly believe that Mike Mearls and team are trying to develop the best D&D they can and certainly wouldn't have any problems with a 3pp extending and expanding their work.
Note 2: The Pathfinder example shows another risk of the OGL model. WotC loses customers when thy change the game, as the existing customer base is serviced by another publisher. In this situation the economic effect turns around. The core producer actually feeds the network at its own expense.