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Unity has a preview package of their new input system as Rewired can be expensive for some devs. I'll recommend going with the preview package and the legacy system causes bugs like the ones you experienced. Brackeys even has a tutorial on the new system

Thank you!! That finally fixed our problem :) Are we allowed to upload this bugfix? Because the submission period is over...

I think small bug fixes and accessibility updates are allowed. Just don't add new game mechanics, levels, whatever. When you update your game, add your update details to your game page so that they know that the update is for.