I don’t think we can expect the visitors to our websites to find and change this setting, just to be able to see the embedded tweets. Most of them will not care, or think our websites are broken. As a result, I expect major sites to move to their own Twitter embed scripts or just stop using Twitter embeds.
I don’t know whether it was a change in Chrome or a change in the Twitter embed script that now causes this exception, but if I were Twitter, I would seriously consider this an issue to take up, and not just push it down to the end-user. You script should not throw an error because localstorage is not available.