Hello Happy smile
Thank you for the question.
According to my interpretation of the GTC, the purpose of travel insurance is to cover unforeseen events, which is why it is recommended that you take out travel insurance at the same time as purchasing your ticket.
According to the following screenshot with the extract from the GTC, in my interpretation it is no longer an unforeseen event and the cover will probably be excluded if the trip starts in the next few days (or has already started) and you already feel ill at the time of taking out the insurance:

Further information such as the coverage information, general terms and conditions including duties in the event of a claim, as well as contact information for Allianz Travel customer support can be found in the following links:
Ticket Insurance Europe:
Ticket Insurance Europe Plus:
If you have any further questions or uncertainties, please contact the official customer support of Allianz Travel, which defines the insurance benefits and general terms and conditions and is the service provider in the event of a claim.
I hope this helps and wish you a speedy recovery so that the trip to Europe and Switzerland can go ahead as planned.
Kind regards,
Nicole