This is the exact situation the US Confederation was in before they became the United States. The solution was that the vote on dropping the veto had to reach a qualified majority that would form a coalition of the willing and move forward without the opposition. In case of the USA the opposing states joined later. We could do the same.

You are right. They want to get rid off unanimous voting. It would be replaced by qualified majority voting (QMV) which the other post refers to.

As a European federalist I can assure you that we need to get rid off the veto. The USA became the United States in the one moment they dropped the veto.

After Brexit no European nation that wants to keep their sovereignty will leave the union. I can highly recommend reading Timothy Snyders „The Road to Unfreedom“ on that topic: The EU is a group of failed empires (FR, ES, IT, PT, DE etc.) and nation states that are too small to exist without being absorbed or controlled by larger empires (especially eastern Europe) if they don’t stay together.

Thus even if the EU becomes more integrated it will the lesser evil.