... with the following option have any serious negative consequences?
Instead of downvoting to below zero, one could suggest a tag for the question that would identify it as a kind of question that the downvoter would like to avoid reading in future. Downvoting a particular question below zero doesn't do anything about future questions that are similar.
If the person who posted the question accepts the tag, then there is a solution for everybody. The only problem that I see is that there could be too many different tags for filtering out what people want to filter out, and that it might be inconvenient for people to include all tags to be avoided if they have to handle the task one at a time, with no assistance from software that can select reasonable estimates of appropriate defaults based on the user's past, recorded activity.