Does falling short prevent one from inviting to Allah, and if so, does that become a justification for closing the door to da‘wah, or is da‘wah obligatory even with shortcomings? And what about the loss of a good example?

Calling to Allah is the work of the Prophets and their followers. It requires those who undertake it to possess essential qualities and necessary manners, the most important of which are: acting upon knowledge and steadfastness in conduct, so that one may be an example for those one calls. Both calling to Allah and steadfastness are obligatory. If steadfastness does not accompany the call, it remains incomplete. A good example has many advantages, including the swift transmission of good to the one being called, and their conviction in the applicability of what you are calling them to.