I can't trace the exact origin of the bug. My assumptions were wrong, or at least not accurate. I am not sure that the suggested changes will help to get rid of the problem 100%. I need a more detailed explanation.

You mean that the bug's happening to you but don't know why? I don't know how reliable and unreliable channels work, like if one is faster than the other or has an order of execution. Maybe MSG_ONE messages get executed first than unreliable ones so that's why is unreliable (check if messsage can be sent, then do it else don't). As PreThink is called by the engine on every Sv_RunCmd, sending MSG_ONE to remove the icon is queued before the MSG_ONE_UNRELIABLE on the removal of the icon. I'm not 100% sure about this, just tried to change the msg_type and the icon always got rid of the HUD.
That's why on the comment I said "fix", maybe it can still happen but haven't to me since that change. Maybe making it reliable can crash some players and kick them for Reliable channel overflowed. The other fix that I thought about is to save the time the player spawned, make like a gap of 0.1 seconds after he spawned and on CBasePlayer::UpdateClientData send this icon if it is not shown, idk

Have you tried instead changing this https://github.com/s1lentq/ReGameDLL_CS/blob/c30fe01f8e60ea1a6a4896bc9e0beb09ae0db580/regamedll/dlls/player.cpp#L10239 to MSG_ONE_UNRELIABLE , maybe they work better when in the same channel?

Yeah, I tested using that channel and sometimes got removed, others didn't or even not showed up the icon