I am trying to figure out a way to best understand how work is distributed across three team members so that when new tickets are created, the manager of the space knows who the best team member to assign it is. The app does not need to determine who is the best next assignee.

Ultimately, the stand-alone 'How many 'in progress', 'paused', and 'blocked' tickets per assignee falls short because one person with few tickets may have a longer Estimated Time to Complete than another assignee with many tickets that take little time to complete. Hence, something like a Gantt chart that takes each of the three assignee's ticket numbers and estimated time and plots them in the calendar format. Is there a built-in app? I tried 'Time Tracking Report', 'Issue Calendar Gadget' but obviously neither accomplish the previous requests.

I see some apps, Issue Calendar for Jira (perhaps), TeamTide – Capacity Planning for Jira (bad reviews; free so questionable), Activity Timeline: Resource Capacity Planning, Time Tracking (incurs extra pricing), but I'm hoping there is perhaps a gadget set-up that may work better than calendar in above picture that I maybe overlooked.
My superiors are also leaning against using Scrums/sprints. A calendar or Other graphical interface that can show the aforementioned data in a calendar format or another helpful graphical interface would be much appreciated. The tickets don't have due dates as each ticket is its own individual task and the effort to close can vary wildly.