Steam game not showing in EA App

Product: The EA app
Platform:PC
Error Report ID (learn how to create an Error Report ID in the sticky post)
Which client functionality are you experiencing this bug with? Launch Game
When did this happen? (dd.mm.yy hh:mm) 1/18/2023
Summarize your bug Do not have access to a game purchased in Steam. Game does not show up in EA App library
How often does the bug occur? Every time (100%)
Steps: How can we find the bug ourselves? I don't know.
What happens when the bug occurs? I get an error message in the EA App stating I do not have access to the game.
What do you expect to see? The game launching

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I purchased a game through Steam in October. I didn't get around to playing the game until this week. When installing, I was prompted to install the EA App, which I did. After installation, I tried launching the game in Steam and received an error message in the EA App stating that I do not have access to the game. I have contacted both EA and Steam support multiple times and both are telling me that I need to contact the other.

The support process has gone like this:

1. Contacted EA Support which told me to reinstall. I still continued to have the problem.

2. EA Support then looked at my account and said the game is not linked and I must contact Steam.

3. Contacted Steam and was told that since they did not create the EA App I must contact EA.

4. Contacted EA Support again. They looked at my account again, said the game is not showing up in my account and I should contact Steam. I explained to support that I have already done this and that they told me to contact EA. 

5. Support then requested I submit a proof of purchase. I could not attach a file to the case that I had opened. EA Support told me that happens sometimes and to close the case and resubmit with the proof of purchase attached. 

6. I opened a new case and attached my proof of purchase. EA Support again said that the game was not linked to my account and that I need to contact Steam. I once again explained that I had already done this and that the last support specialist told me to submit proof of purchase in a new case. EA Support ignored this and said that there is nothing they can do.

My EA and Steam accounts are linked so I know that there isn't an issue of purchasing the game on the wrong account.