In version 1.16.5 of Minecraft on a private server I am paying for using Serverminer.com, I have found that the gravestone is very unreliable. What the mod page does not tell you is that there are two options for gravestones that you can drop, one being the aforementioned Grave and the other being a floating totem that does not spawn on a block, but instead spawns as an entity. I call it a totem, but I think it's meant to be the "friendly ghost" the mod offers. I only call it a totem because it reminds me of the Totem of Undying in the base game.

This totem spawns only spawns AFTER your first death without having claimed your items from the initial grave if you are to die a second time. Breaking both results in you dropping said totem after you die from that point instead of a grave if you were to die again. If you were to do the same thing (I.E, die, drop a totem, then die again and drop a grave before claiming the items in the totem) the totem will then despawn and drop all of the items that it contained, as if the player died in the base game, and then the player will drop a grave from that point on if both the totem and grave were claimed.

I ran an experiment on this if you were to do these in order:

1. Die and Drop a grave
   

2. Die and Drop a totem
   

3. Die a third time before claiming both the totem and the grave
   

I found that the mod alternates between the Grave and the Totem for no reason and I cannot find any option to fix this in the config file "gravestone-server.toml"

You will find exactly how I have it configured here.

#If this is set to true you get an obituary after you died
enable_obituary = true
#The blocks that can be replaced with a grave
#If it starts with '#' it is a tag
replaceable_blocks = ["#gravestone:grave_replaceable"]
#If this is set to true the obituary will be taken out of your inventory when you break the grave
remove_obituary = true
#If this is set to true only the player that owns the grave and admins can break the grave
only_owners_can_break = false
#If this is set to true the ghost of the dead player will be spawned when the grave is broken
spawn_ghost = false
#If this is set to true the ghost player will defend the player
friendly_ghost = false
#If this is set to true you get your items back into your inventory by sneaking on the grave
sneak_pickup = false
#If this is set to true you get your items sorted back into your inventory by breaking the grave
break_pickup = true

Another weird issue is that I used this mod on my singleplayer world and have not ran into this issue at all whatsoever. No matter how much I die, I continually drop a grave, even if I die before I can break the grave. Am I missing something? Is this a server exclusive issue? I can't find anything like this on the FAQ either.

You can't use two mods that process deaths at the same time.