The TLDR of the replaysystem is, that it records network traffic and replays it. Therefore, only actions, which are replicated, are recorded and part of the replay. I doubt the template was build with Networking/Replication in mind, therefore nothing but the player is recorded. After all, the unreal character has some solid replication build in, which is why it is recorded.
We use it in our own project, and the way it works is really cool. Because our game is Multiplayer, we get it out of the box (mostly )
What you can try for your template based project (result quality may vary!):
Set the boxes to replicated and tick “replicate movement”. Go into the character blueprint (hope its blueprint, because its easier then for an example) and go whereever it triggers the fire logic. If the logic is a seperate function, mark this function as “executes on owning client”.
These are just first steps, and a more in depth rework of the template might be required. IIRC there is a whole demo multiplayer shooter somewhere as an example project by epic. AFAIK you can get it through the launcher.