I think Epic had a chance. Valve have been licking their wounds on the failure of the Steam Box, Steam Controller and Steam Link for years. Their user experience has felt like it was lacking for a long time, they made a big bet on PC VR and it's looking like a busted flush right now (I say right now because I think Valve will turn it around), and there was growing disquiet in the developer community about the way Valve was running its store. There were people saying Valve needed a kick up the arse or someone else would take their place for some time.
Epic clearly saw an opportunity to disrupt, but they made dumb decisions from day 1 and while Epic fannied about with keeping games off Steam (and spending exorbitant amounts of money to do so) Valve was working on streamlining their UI, developing the Steam Deck and really creating a clarity of vision about what the company's destiny would be.
As a result, Valve is once again reshaping the PC gaming platform and Epic feels like they're competing with the version of Steam that existed 15 years ago.
Epic clarely was workign on whatever EGS became since well before EGS announcement (as Sergei was working with "other company" already by like 2017), so a lot of the media and indie build up around was clearly influenced by them to make it easier to make inroads. The Valve UI change thing is a bit weird to point out because Chat (the first part of it) launched way before EGS), and Library a bit afterwards. I would say that EGS did make them be more vocal about the upgrades tho.
Epic strategy in general was have a blitz in 2019-2020 that shattered the idea that "non F2P PC Gaming is Steam" hence why they paid a fuck ton of money to remove Metro Exodus out of Steam weeks before launch, the Ubisoft deal, and the RDR2 1 month exclusivity. All of those dont really make that much sense (the Exodus one even less) if we do not consider that Epic wanted to disrupt Steam image to allow a competitor. You remove all those games (and EA and MS) and suddenly none of the big AAA games of those years would have "come to Steam" really.
Once you disrupt the idea of Steam being "the place to be for PC gaming" a ton of the first mover advantage would be gone, and Epic could poach a ton of the big guys which are the ones that bring the most of the "base growth", which is what most people work around. Epic would then use that platform to upsell their games and make more money that way. Indies would then have to choose between easily getting on a Steam that lacks some of the big hitters that would impact Steam's growth or visibility, or ride the Epic wave. If fully succesful, Steam growth would have stopped (and maybe reversed) and Epci could have gotten the 30-50% of the market if everything went well. (I doubt it would have happened, Steam still had a ton of momentum and strength even without AAA games).
The problem was that Epic completely misread the timing of the situation, they kinda came just as there was a giant sea change already happening in how big gaming companies saw platforms. Preivously, big companies tried to maximize profit by maximizing the cut htey perceive, so from the same amount of money, they make more. As service started to become more prominent (and through that lenses it is not that rare that EA and MS, the ones with subs came back first bakc to Steam), the way to maximize profit is to increase revenue by being everywhere, as that brings more eyes into your product and more people to play them, increasing chances to get whales.
If Epic had come one year earlier, their strategy would have been more succesful (as the service sea change was moving slowly still, consider EA didnt really have Apex), but Epic also didnt have as much money and were unsure how to fight the mobile markets (which were the big end goal). Epic aggressivity, whose main goal was shaterring the perception of Steam as PC Gaming ended up backfiring when they couldn't shatter that perception, so in the ned they looked like they were doing it out of spite. Pure "if you come to the king, you better not miss" (they missed)
What makes it more funny is that you can see Epic is still focused on the "maximize profit by not giving any money" by how they treat mobile. Mobile is a huge market but Epic refused to put their game in Google Play (only doing it to sue Google later on), making them lose a ton of Android money in general (and making PUBG Mobile even more of a strangehold in mobile than it should be). So it is not just something they missed in PC. Like they could still be their own platform without needing to keep the 100%.
Personally I would say I also didnt expect MS and EA to come back to Steam, both because I also didnt see hte change to service focus (until it happened lol) and because of "sunken cost fallacy". So I cannot really blame Epic for missing it. I can blame them for keeping up some of the strategy even once it was clear it was bust when MS came back to Steam (asking MS to become Epic co-exclusive was idiotic and a way they gave free exclusives to Steam because it was clear MS cared less about the money and more about intangibles with that move.).