I have opened a ticket with Power BI support, and this is their response:
We have checked with our product team on your issue and confirmed as below
Usually the service would take more time when you download the report. But as our issue it is taking more than expected.
Product team is confirmed that it is expected behavior when downloading PBIX file from service.
When downloading the PBIX, the Power BI service has to rebuild the PBIX from scratch again in such a way as not to use too much resource (so not impact other processes).
This is why you don’t get the ‘save as’ dialog until the end as this only prompts when the file is built and ready for download.
All in all, this does take a lot of time and the larger PBIX you have the more painful the experience.
Feedback has previously been sent to the Product Team about this and they do intend to look into the process and how it can be improved, however it’s not going to imminently change. There is no ETA from Product team.
It seems like we have to wait for unknown time to get this fixed...