My wild ass guess is that Contact has a required standard field that's can't be written to.
And to find the exact field - put the logic into a unit test.
Start with a method that has one runas user - the customer community user.
Duplicate the problem.
Then fix that with a method that has two runas users - one with a customer community profile - and another ...
Has per @KrisGoncalves's comment, I updated the Sharing Set to allow Read/Write for the selected Account and it started to work.
I realized why read/write sharing is required for Account. Since Account is the parent (in sharing terms) for Contact and Contact is the master details parent for Ticket__c, I actually have to have write sharing on Account ...
I found a solution by using .comm-navigation__item .slds-container_fluid it's only working if I use .slds-container_fluid.
I wonder whether this is a solid solution? Any recommendations, would you suggest to go that way?