Providing Users access to MY files via a Dropbox API app.. I don't care about their files

You're correct that the API was designed with the intention that each user would link their own Dropbox account, in order to interact with their own files. However, it is technically possible to connect to just one account. The SDKs don't offer explicit support for it and we don't recommend doing so, for various technical and security reasons. For web apps though, most of these concerns are allayed, since you can protect your access token server-side.

If you did want to go this route, instead of kicking off the authorization flow, you would manually use an existing access token for your app and account. (Just be careful not to revoke it, e.g. via https://www.dropbox.com/account/security .) That is to say, you just need to process the app authorization flow (or get a generated access token via the App Console) for your own account once and then hard code that access token into your app for use when making the necessary API calls.

Alternatively, you could just use a shared link to the folder you want to share.