The question asks "I would like to know if there are any official recommendations on if this kind of travel is discouraged/forbidden, or even unofficial reasons why one option might be better than the other."
In the US, travel and similar restrictions are being handled mainly by state and local governments. Chicago is in Illinois.
The State of Illinois has prohibited non-essential travel. See EXECUTIVE ORDER IN RESPONSE TO COVID-19 (COVID-19 EXECUTIVE ORDER NO. 8).
"All travel, including, but not limited to, travel by automobile, motorcycle, scooter, bicycle, train, plane, or public transit, except Essential Travel and Essential Activities as defined herein, is prohibited."
The various essential activities listed in the order do not seem to me to cover leaving Chicago to stay with relatives or to avoid potential civil unrest.