For the objective of posting photos to an Instagram account while preserving as much privacy as possible, your approach of a separate machine and uploading using its web browser should be sufficient. That said, Instagram for web could also be sandboxed using a private browsing tab on your existing desktop. Certainly, avoiding an installed app – such as the mobile app – will prevent the most obtuse forms of espionage/tracking.

That said, your titular question was about how to maintain an Instagram account, not just post images. And I would say that as a social media platform, this would include engagement with other accounts and with comments. For that objective, having a separate machine is more unwieldy. But even using a private browsing tab on your existing machine is still subject to the limits that Instagram intentionally imposes on their desktop app: they save all the crucial value-add features for the mobile app, where their privacy invasion is greatest.

To use Instagram in the least obtuse manner means to play the game by their rules, which isn’t really compatible with privacy preservation. To that end, if you did want a full Instagram experience, I would suggest getting a separate, cheap mobile phone (aka a “YOLO phone”) to dedicate to this task. If IG doesn’t need a mobile number, then you won’t even need a working SIM account. Then load your intended images using USB file transfer, and use an app like Imagepipe (available on F-Droid) to strip image metadata.