I have a similar problem like a couple people before, I don't see a private app development option. I read that the reason is that I don't have access. But I'm the founder of the shop, no one else. I'm an admin and this option is still not available.
Private App is no longer available, you can use Custom App instead.
Private apps are deprecated and can't be created as of January 2022. Ask your app developer to create a custom app. Like private apps, custom apps are built exclusively for your shop, but they don't require open API access to your store or access to your Shopify admin.