Thought I could edit my last post in time to add this. Here's some relevant info from the output_log.txt. It appears once right before I join the server and only when I go to join.
Fallback handler could not load library /srv/sevendaysd/server/7DaysToDie_Data/Mono/x86/SteamworksNative
Fallback handler could not load library /srv/sevendaysd/server/7DaysToDie_Data/Mono/x86/SteamworksNative
Fallback handler could not load library /srv/sevendaysd/server/7DaysToDie_Data/Mono/x86/SteamworksNative
Fallback handler could not load library /srv/sevendaysd/server/7DaysToDie_Data/Mono/x86/SteamworksNative
Exception: ServicesGameServer.Initialize failed: SteamInitializeFailed
at ManagedSteam.ServicesGameServer.Startup (UInt32 ip, UInt16 steamPort, UInt16 gamePort, UInt16 queryPort, ServerMode serverMode, System.String versionString) [0x00000] in <filename unknown>:0
at ManagedSteam.ServicesGameServer.Initialize (System.Globalization.CultureInfo activeCulture, UInt32 ip, UInt16 steamPort, UInt16 gamePort, UInt16 queryPort, ServerMode serverMode, System.String versionString) [0x00000] in <filename unknown>:0
at ManagedSteam.ServicesGameServer.Initialize (UInt32 ip, UInt16 steamPort, UInt16 gamePort, UInt16 queryPort, ServerMode serverMode, System.String versionString) [0x00000] in <filename unknown>:0
at SteamMasterServer.Connect (UnityEngine.MonoBehaviour _mono) [0x00000] in <filename unknown>:0
at GameManager.ᢵ▒~Y~C () [0x00000] in <filename unknown>:0
at GameManager.Awake () [0x00000] in <filename unknown>:0
EDIT:
Solved, I think...
I think it has to do with running multiple instances of the server. I had left an instance running from when I did a vanilla install following Alloc's guide. Once I stopped that, I could connect to my server that wasn't using the management scripts.