You can’t pass in an individual field value as a context parameter. In the example you’ve presented, you’ve defined an Issue Field called “Client Email”. JIRA doesn’t currently implicitly consider the issue context when it loads your web panel.
What I would recommend is you use the issue.id context parameter, like "url": "/profile/{issue.id}" and then make a client-side request for the field value from your add-on using AP.request to a URL like /rest/api/2/issue/{issue.id}?fields={addonkey__email}.
Make sense?
Is a change to this behavior being considered? I’m dealing with a similar situation involving entity parameters. My add-on is a simple integration with an external service and currently is only using the Atlassian Connect JSON, so working around this would add considerable complexity.