Entering through the quick-add is the most common way to add contact information. The issue you are running into, is, your user permissions call for your user to not be able to see or set the owner, and Infusionsoft does not currently assign owner based on who created the contact, automatically.
I have an idea that would require a simple internal form, set as your ‘Default’ quick add form. This form could contain a required radio option of your users, requiring your users to just select their name when entering new contact information. This form could run through a simple campaign to assign the owner.
I have created a video, visually showing what this would look like. I laid out an internal form, and 3 user sequences (my example accounts for 3 different users, but could easily accommodate more.) https://www.screencast.com/t/TmghmjESggo
You will see in the video that I create an internal form, and along with any information fields, I include a ‘Radio Snippet’ that lists out all of my users, and I set it to *Required.
I continue the campaign by setting up a decision diamond to direct my contact into a specific sequence, based on the radio button selection, on the internal form. Inside the sequences, I just add a ‘Set Owner’ sequence, configured to the specific owner that should be assigned, based on my rules that passed the contact into a specific sequence.
Once I publish my campaign, I navigate to my CRM Settings, where I have the option, at the bottom of the settings page, to set my new internal form as an available ‘Quick-Add’ form, and, I even set it as the default in the video. If you set it as the default, it will be the form that displays when you hit the + at the top-right of the screen, to add a contact.