Exporting Files
Apps like figma and sketch, have a small export section at the sidebar. I'm not sure if they provide an export dialog with finer finer control options.
Prior Art
Affinity Photo Export Implementation
This is the export dialog in Affinity Photo

Also affinity photo has a dedicated export persona

Glimpse-NX
This is the quick export popup design from the Glimpse-NX project:

Pixelmator Pro

I'm not quite sure what is the best way to export. I like the idea of having a small export dialog like Pixelmator Pro with an option for displaying more advanced options for finer control. Also, if we need to decide if we need to implement an export persona or not.
PS These are initial thoughts and don't think my opinion is rock solid. It is a subject to change.
I think "fast" defaults with the option to choose more advanced options is the best (which is basically what you suggested)