Does hosted on Itch mean there is a web build required? Or just that the downloadable executables must be hosted on Itch?

yea no downloaded stuff are allowed. I just mean it has to be able to be downloaded on the itchpage

It means that in case you choose to export your game as a downloadable .exe file (instead of choosing the embedded html ) you should directly upload the file on itch.io. They will not allow you to upload the file on other sites / storage solutions and only provide a link to that 3rd party site. This is due to security reasons.