"Does my organisation need a cloud storage account like Dropbox or OneDrive or an organisation storage tool like SharePoint?"
It’s a short answer.
If you don’t want to share, manage anything or change the way you’ve worked since you first used a computer then use Dropbox, Google Drive or OneDrive. Otherwise go to something that allows you to store things that others can find. It’s as simple as that.
Cloud storage accounts are a good replacement for your file server, if that is all you want to do. You can synchronise your local device drives with remote folders and now you have files backed up off your device.
"But I want others to use these files".
That is a different proposition. I’ve recently seen file server folder structures that have been set up with great intentions by multiple people who have since left the organisation. Good luck finding anything because you must think like them to recognise a naming convention or structure.
One organisation scheme for folders that I’ve seen requires filenames to be prefixed by their folder paths, which led to very long file names. If you are confident everyone will think like you, when you set up folder structures, then continue with a cloud storage account for your organisation.
If you want to avoid the difficulty in finding files, then you need a tool designed for collaboration and sharing content (content is a collective name for documents, video, images, social or news media feeds). Microsoft SharePoint is a great tool designed for organisations to collaborate. Google Docs has some of those features too, but as seen recently this can come with unwelcome issues, and as everyone knows, nothing in this world is free. If you’re not paying for it, you’re not the customer, you’re the product.
They are more than just file storage but tools to allow you to find content easily - perhaps create a team’s own area, with content organised just for you. For example you might have a special project underway and you want to people to be informed by shared calendars or Facebook feeds, as well as being a place where all the project documents can be found (easily!).
If you don’t want to make it easy for others to find content, then continue to work the way you always have.