Creating a Gantt Chart with multiple circles is definitely possible, however you approach may seem not to be the correct one. You should create a Gantt Chart first and then you can use a dual axis for the points in your different dates extracted with calculations.
My team created this for a client last week using Tableau.
If you want, you can upload your workbook as a TWBX and we may be able to help you in more detail.
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Thanks for the quick response!
Yes the screenshot you attached is pretty much what I'm trying to achieve. Unfortunately I cannot share a workbook as the data is confidential.
I assume the way you create the circle marks is by having a variable with different dates in it for the same level of detail? The problem is that my data has rows with 3 specific date variables (ie startdate, mid, stopdate). IMO this implies my data is too flat for what you are showing, because I cannot aggregate these into one variable?