I understand the privacy concerns regarding a digital record of vaccination. But it's not really that much more private unless you somehow got the vaccine without the Gov't knowing who you were.
If private businesses want to require proof of vaccination, they should have the right to do so, same as if you chose not to get a vaccine. Similar to a no shoes, no shirt, no service policy. Every action has consequences, and people who complain that it is "discriminatory" just don't want to face the consequences of their own actions.