@qphawk There might be multiple ways to achieve this. Try below suggestions:
Using Power automate flow:
Create a simple power automate flow which will run on item creation/update (trigger can be as per your requirements).
Add "Update item" action in flow to update the Date Assigned column using expression: formatDateTime(utcNow(),'yyyy-MM-dd')
Using Power apps:
Customize list form using power apps.
In OnChange event of Person Assigned field, update the value of Date Assigned column using variables
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