This article is an interesting piece written by a group of reenactors and HEMA-practitioners in France. Their main argument is that in the later 12th and early 13th century, historical artwork suggests that shields did not have a strap for the forearm. Instead, they argue, the shield would be held by pairing the hand strap and part of the guige together. They provide numerous historical images to support their argument.

Read more, here: https://www.degueulesetdargent.fr/2017/03/02/shield-straps-and-holding-of-the-shield-in-the-early-13th-century/

This looks to be an excellent article. I have not read through as yet but I see some very good info here.

Craig