They almost did with the 360.
Sony dropped the ball massively, but changed a lot mid gen, so even if the XBO was a perfect copy of the 360 it would still lose to the PS4, as the 360 was already losing to the PS3.

So yeah, the 360 gen, but only that one.

That was the only gen I ever considered picking anything else but a Playstation, as I liked Sony games way more I still went for the PS3, but the 360 was a very decent choice as well.

Actually I'm debating on finally getting one this last few days, I wanted the original Xbox instead (I really like the X shaped top), but getting a good working unit that comes with the box around here looks more unrealistic by the day, it's absurdly pricey, so I guess I could just get the 360 to play Xbox games with BC and also finally be able to try Lost Odyssey.

7th gen. If they didn't have the PR clusterfluff that was the Red Ring of Death which was an absolute nightmare...
And despite the Wii selling like hotcakes... It only sold 17~ million units more than the Xbox 360.

It was just hard to beat Halo, Gears and Forza's 1-2 punch that generation, which helped to push back the RROD PR disaster.

In hindsight, the Xbox 360 was the best console of that generation, it had the best performing multiplats, it had the best graphics, it had great support.
It was let down by the RROD.

Speaking of RROD, I'm actually wondering, did this harm sales, or actually boost them due to all those people buying a 2nd/3rd/4th 360 after hardware failures?

No, but if there was ever a chance, it was the 8th gen. The 7th gen would have required Nintendo to not succeed, but Nintendo did, so that's the end of that story. They could have beaten PS3 with more investment into Europe, but I don't think beating Wii was up to Microsoft (realistically). The 8th gen on the other hand was following the 360's success, so with some actually good moves instead of screwing up right at the announcement (or before, considering the announcement was just a follow-up to everything that occurred before it), there's at least the benefit of doubt. Sony didn't screw up that generation however, so I don't really think Microsoft could actually have won, but hey, benefit of doubt!

I don't have any hard data but I remember a lot of people saying they were holding off 360 until the Jasper chip came out which promised to drastically reduce RROD. Then again working at Gamestop we gave the pitch of the 3 year warranty. We sold far far far far more 360's than PS3's in those 2 years as a manager. It's also easy to forget tho PS3 had a high failure rate with YLOD but wasn't as publicized.   

that's a big fat no! you can't even make the case for the 360 with PS3 aligned sells. Simply put, it was never possible.

Not unless PlayStation bungled the EU badly during the 7th generation and the early 360s didn't have such high failure rates.

Under such dire circumstances maybe the 360 could have beat the ps3 but probably not the wii

Probably not, simply due to differences in system preferences from region to region. Xbox always struggled in Japan, and while the 360 saw Xbox expand its market share in Europe, that region still greatly preferred the PS3 (well, except the UK, which more closely resembled the U.S.). The 360 dominated the PS3 in North America, though, and eventually became the #1 home console of that generation and is still the #2 home console of all time in the U.S. (behind only the PS2) and #4 system of any kind (behind only the DS, Switch, and PS2). Because of the way the regional market shares worked out, the PS3 still managed to beat the 360 worldwide despite the latter having one more year on the market and a solid price advantage for almost three years into the PS3's life. The 360 was MS's best shot, and they couldn't quite beat PlayStation. There were all sorts of variables at play, and it's hard to say if one or two more of those went MS's way that the 360 would have come out on top, but even if it did maintain a narrow lead over the PS3 worldwide, there was still the regional preferences to take into account, plus a high likelihood that the PS3 still managed a better per-year average.

Going into the following generation, MS really screwed up the lead up to the Xbox One. There were the announced anti-used games measures, force-bundling of Kinect, mandatory online for the system to work (and also Adam Orth and Don Mattrick putting their feet in their mouths on that issue), and a reveal event summed up as "TV, TV, TV, sports, sports, TV, TV, Call of Duty." While MS did backpedal on those anti-consumer measures, it was too little too late, and Xbox lost a tremendous amount of market share, arguably at the worst time to do so.

But even if the Xbox One had none of those issues and was just a solid $400 console that didn't really change the status quo for used games & online and left Kinect as a fully optional accessory, inertia is a cruel mistress, and it's entirely possible that the PS4 would still have won. Seeing as both systems released at the same time, I still think the PS4 would have sold better worldwide each year even if the XBO continued the Xbox brand's domination of North America, simply because the PS4 would be the overwhelming favorite in Europe, and Xbox would still be a non-starter in Japan. The gap would have been much narrower than it was in our timeline, but I still think the outcome would still have been the PS4 winning by several million units.

Maybe if MS had, in addition to not botching the XBO, had also put a ton of effort into at least trying to match PlayStation's sales in Europe since at least Gen 7, they could have possibly continued to narrowly defeat PlayStation every generation to date. Japan was probably not even worth it, as even with a bunch of exclusive JRPGs Japanese audiences simply had no interest in hardware that wasn't from a Japanese company. But Europe could have been closer. Brand loyalties may be strong, but they're not always absolute, and the 360 having a major price advantage helped demonstrate that a large portion European gamers were willing to shift, and perhaps even more of them may have if MS put in more effort.

Without RROD, the 360 would have won the generation. It would surpassed the Wii and PS3 sales would been significantly lower. Even with RROD, the PS3 didn't out sell the 360 until after the PS4 and XB1 had launched. RROD is what saved the PS3, without, it never would have recovered the way it did.

The X360 without RROD might have beaten the Wii.