Well, someone suggested on the issue tracking thread using the ESPN app on a smart TV, but I don't know if you have a secondary smart TV at your house that is not a Roku TV that is capable of running the ESPN app. If so, I remember more than one person was pretty happy with that workaround. They said using their native Smart TV app downloaded from the TV app store was pretty much the same process as authenticating with the ESPN app on Roku, which you already seem to know how to do. I couldn't give you a step-by-step of that since I don't have a Smart TV.
If you want to try to use the website, I can try to illustrate with screenshots with a step-by-step. I only have ESPN3.
https://www.espn.com/watch/
Technically, you don't have to even be logged into your Disney account for this to work if you only want to watch ESPN. It just won't save your progress on whatever replay you're watching if you're not logged into it. So, if you go to the website and click on the profile icon in the upper right corner, there is a tv provider option. Click on that.
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From there, click on Verizon and it will take you to the Fios login. Login with your tv provider account.
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It should take you back to the ESPN watch webpage. If it authenticated correctly, you should have a logo in the right corner like I have. Mine is AT&T U-verse.
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From there, you should be able to click on a replay and see if you have access to it. If you don't have access to it, it should give you this message.
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Here is a replay that shows it playing (my ESPN3). It will show your tv provider logo in the right corner.
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If it says get TV provider, it means you're not logged in to the TV provider. Any other messages besides that are most likely going to be error messages. If you get any error messages and you plan on talking to ESPN, maybe they can help you with the website at least if you tell them what those error messages are. If you can get the website working, and you have the ability to cast to your TV from your device, you might be able to watch it on your Roku TV. I don't have a device that can cast ESPN to my Roku, so I can't give you much advice on that. Sorry it's not a solution for the Roku issue. 😞
Have a good evening! 🙂