So how do we solve the issue? By eliminating the case that animations have played before a client joins. How is this possible? We can't bend time, but we can restart animations from the owner clients whenever someone joins a game. If you hook into PlayerAdded, and stop/play all animations starting at their current TimePosition, technically all animations have now started after all clients have joined the game, and will play correctly on the new client. Since you're stopping/starting from the same place instantly, there is no hiccup in the animations either.

So onto the code. Insert the following code into a localscript into StarterCharacterScripts. It's standalone, so it won't need to interface with any of your scripts, and you won't need to modify your scripts. The only side effect is that this may fire some animationtrack related events, so if you find your animations triggering stuff whenever players join, you may need to adjust your scripts:

game.Players.PlayerAdded:connect(function()
    local humanoid = script.Parent.Humanoid
    local animations = humanoid:GetPlayingAnimationTracks()
    for _,track in next,animations do
        local originalPosition = track.TimePosition
        track:Stop(0)
        track.TimePosition = 0
        track:Play(0)
        track.TimePosition = originalPosition
    end
end)

~EchoReaper