The answer depends on your definition of "efficient". I will give you the prons and cons of each methodology below so that you can choose for yourself:

1. FTP: Using the FTP automation task to edit specific php pages in wordpress?

This will not work. The php pages do not store static data/information in them. They just contain the "code" that is executed on the server side and is essentially responsible for saving the information that user enters (e.g. a post) into the database. This practically means that the information that is stored in a wordpress site, is not actually stored in the php pages but in a database (in most cases it's a mySQL database).
Pros: Not applicable (not valid method to post in a Wordpress site)
Cons: Not applicable (not valid method to post in a Wordpress site)

2. DOWNLOAD FROM WEB: 

The form that you would have to fill through the Download from Web action in the post-new.php of a WordPress site, contains a ton of inputs that need to be filled. Most of them of course are hidden and have a standard value, but in order to automate such a huge form, a lot of regular expressions would be needed and a lot of time would have to be spent. Moreover, if for some reason wordpress changes the input fields, or something else is added due to a plugin for example (I'm not sure if this is possible but I'm guessing based on my experience from other CMS) you will have to study the whole job again and try to figure out what they changed. 
Pros: Very reliable, works in the background, as fast as it gets, can be transferred to other machines as a compiled job.
Cons: A lot of time needs to be spent to build the job, requires very technical knowledge (such as regular expressions), hard to detect the error if for some reason the submit form changes.

The XML-RPC remote publishing protocol seems very interesting. I do not have time to test it right now, but according to this post, its author claims that he was able to build a very simple html form to post to a wordpress site. If this is possible, a simple way that I can think of to automate this task through the "Download from Web" action, is to setup a very simple WAMP server locally, a very basic html form with the fields that will be different from post to post (e.g. title, body etc), and fill this form with the "Download from Web" action.
Pros: Reliable, works in the background, you can create a very simple html form so that the download from web action becomes very simple, is not error prone if the form is changed.
Cons: The initial configuration may take time and requires a certain amount of knowledge about XML-RPC, takes a lot to build your job and all the required html forms etc.

3. Image Recognition:

Through the "Wait for Image", "Move Mouse to Image" and "Send Mouse Click" actions.
Pros: Reliable results, easy to detect problems if the interface changes
Cons: Takes over the computer, may be slow depending on the images that you capture, depends on the interface (not the code) so if the theme of the website changes the image recognition actions will fail, will need an autologin solution if you will have to run it based on a schedule when logged out, faster to create it than the download from web methods but slower than the macro recorder.

4. Macro Recorder (Coordinate Based):

Through the "Record Macro" option in the designer.
Pros: Better execution speed than the Image recognition, you can have the process automated in a matter of minutes with macro recorder (very fast in terms of producing the job).
Cons: The page load times may vary so it is very unreliable if not used in conjunction with the image recognition system. If new toolbars are introduced to the browser, or anything that may change the position of the various elements, the job will fail.