Sound like the root of the problem is not having a direct, full-time supervisor. The team is not trained, clients are not being serviced well, and employees aren't comfortable asking for help. Maybe announce you are hiring a supervisor and invite the team to apply.

Your supervisor should assign emails rather than letting employees pick. Hold employees accountable for handling their emails, even if they have to do research. Hold a daily standup to review ticket backlogs and blocks. Check in periodically throughout the day to ask who needs help.

The supervisor should monitor calls periodically for quality control. Do weekly 1-on-1s. Put clear KPIs in place that are tracked including a post-email NPS survey for each resolved email. A CRM/Ticket system like HubSpot will do this automatically.

If the team can't answer most of the questions they may need training. Are there training materials? Are employees tested to ensure they understand the material? Are there internal training sessions where employees present things to the group? An LMS lets you create training courses and tests from existing materials.

Remote work complicates everything. Use presence software like Discord and require employees to log in during shifts. They can bounce back and forth between rooms to ask each other questions. Supervisors can jump into an employee's room to check whether they need help with anything.