Verdict:

When you read that Battleship Potemkin is a propaganda movie, please be aware that this does not mean that the movie just has a revolutionary tone or a political message, like Birth of a Nation. No, it very literally is a propaganda film where one side is getting viciously vilified and the other side is being wholly victimized. There is a very clear motivational message.
This is different from your average epic that has a clear villain and a clear protagonist. Such exaggeration of characters is normally done for the viewer’s benefit because its gives the viewer to reason to root for the main hero and to be happy when that hero succeeds. In propaganda films it is not for the viewer’s benefit and you can feel that! Propaganda films feel preachy and pushy as their purpose is not entertainment, even though they may be entertaining in the process. Such films therefore have a disgusting core, and unfortunately, for all its epic feel and cinematic genious, Battleship Potemkin is no different.
It is not just propaganda but very ludicrous propaganda and on occasion misses its mark or is hilariously corny. A grown man gets slapped on the back a couple of times in his sleep and that gets him crying. It just feels awkward, like the movie is trying to look for trouble everywhere, clutching at straws.
Other times, it just does not explain things well. There is a scene where the men look at an officer with contempt but does not give a reason for it. I later realised it was because the sailors were refusing to eat.
Visually Battleship Potemkin is very impressive. A few confusing scenes aside, it is very good at conveying its propaganda with simple to understand messages and symbolism. It is also good at creating tension, surprise, atmosphere and just showing off some pretty maritime scenery. There are a few interesting special effects and cool camera movements during the famous stairway sequence.
However, as already mentioned, it goes overboard with the its propaganda. There are other oddities too, like a gun clearly not pointing at a person and yet the person falls to the ground anyway after the gun is fired. There are also several scenes that drag on for too long, with nothing important happening, and that in a movie that is only about an hour long. If you cut out all the unnecessary bits, the movie would probably not be more than 30 minutes. It only has 3 events to show (divided into 5 parts).
The best thing about this film was the music. It was not just providing the mood but also the sound effects for what was happening on screen, which was not a standard thing for silent movies at the time, and the music managed to do so without losing its flow and without stopping the music. The score has a bombastic modern feel to it of a Hans Zimmer composition but a timeless sensibility of classical music.
Now, I should probably point out that, if you see the modern version of the movie, you will likely get the remastered version of the soundtrack. No doubt the original quality would have been lower.
The most pleasant thing about this film is its presentation, consisting of an epic and lively soundtrack and visuals. But its story is presented awkwardly, with the political pushing being painfully obvious and with other awkwardness in the screenplay. This cannot be ignored because it directly affects what you see.