There are many “plates” to “spin” when learning a language:
Range of linguistic structures
Comprehension (listening and reading)
This list is not complete and it is not necessarily in order of importance.
The question depends, of course, on your reasons for wanting to learn in the first place.
If you are a spy, who needs to pass for a native speaker, or indeed a child, desperate to fit in with your new school-mates, pronunciation will probably be at the top of the list. Children have a clear advantage over adults when it comes to mastering the new sounds of a language. That’s official. It is possibly because they are not so conscious about all the other aspects of learning a language (“Should I be using the present simple or the present continuous here?”) and therefore have more space in their learning faculties to devote to good pronunciation. That’s not official.
Of the many non-native speakers who pass through the doors (or these days, the Zoom screen) of Same-Sky, those who are most impressive are not necessarily those with the best pronunciation, but those who are able to use interesting turns of phrase, and relevant cultural references naturally during conversation. A parrot can be trained to have perfect pronunciation, but trying asking it what it thinks about the localised easing of lockdown restrictions.
A good accent is what many students most desire, but it’s almost impossible to pass as a native speaker if you have settled in the country after the age of about ten, so it is a better use of time to listen and read as widely as possible in order to improve your vocabulary. Your accent will improve naturally, the more time you spend in conversation with native speakers – not just your teacher, who has been trained to speak in a moderated way so that everyone understands instructions!
So, whilst the lazy, obvious, but nonetheless true answer to the question is that they are all important, in the humble opinion of the Same-Sky teachers, the aspect to focus on above all others is the desire to learn and use a wide range of up-to-date, idiomatic language.
To put the original question another way: Would you rather spend an evening with a well-spoken parrot or a person with original ideas, expressed in an interesting way... oh, and an endearing, exotic, dare I say cute accent?