Whoever is LOD has no obligation what-so-ever to help out. If they choose to do so, they can. Whoever is LOD is the highest authority in the building. 99% of what they are supposed to be doing is controlling, organizaing, planning, and leading. Notice that none of those things requires helping out or getting their hands dirty.
The LOD is the sole authority in the building on who should be doing certain tasks and what tasks should be done. If the LOD decides that you need to have a 45 minute huddle, then that is what happens. If the LOD decides that instead of working stray/zone at 8PM everyone on the floor needs to build end caps, then that is what happens. If the LOD decides he/she needs to sit in clerical and tell people what to do on the walkie, then that is what happens. If the LOD decides he/she needs to work stray/zone, then that is what happens. Basically whatever they decide needs to be done is what is done. Anyone questioning the LOD is subject to corrective action up to and including termination.
It sounds like you are trying to find out if the LOD is required to help out so you can make an issue of it at your store. (the LOD has no requirement to help TMs work) More than likely, someone on here (who doesn't know better) is going to tell you that the LOD is required to be helping out. If they do, my advice to you is not to go and tell the LOD they are supposed to be helping out at your store. Trust me, it won't end well for you.
"What is the LOD's role as far as helping the team as a whole get the store in shape as we get ready to close for the day"
So basically in answer to this question - the LODs role in the closing shift is to make sure everyone is doing what they were instructed to do, to decide who needs to do what, to decide what needs to be done, walking the store so they can write a report for the opening LOD, and to follow proper closing procedures. (lock the doors, etc.) There is a little more to it - but basically during the closing shift the LOD is 100% management focused.
Now as far as TLs go? On the closing shift they should be *both* helping out *and* leading. Sometimes TMs get the terms LOD, ETL, and TL confused. So if your TLs are not helping out and leading, then there is a problem. However, as a TM that is not for you to deal with. It is up to the LOD to deal with if a TL is not doing what they are supposed to do on a closing shift. Now, if the LOD chooses to let a TL run around and do nothing but "lead" on the closing shift, then there is nothing you can do about it. The LOD is the sole authority in the building - and if that is what they want a TL to do, then there is no debating it.
Technically, if the LOD told a TL to hang out in food ave the entire closing shift and told the TL to sit on the walkie and tell people what to do the entire closing shift, there is nothing you can do about it. Now, I seriously doubt any LOD would do that.... (and the STL would probably question that ETLs judgement ability), but they would have the authority to do that.
Basically the point I am trying to make is that whoever is LOD has basically unlimited power to direct store operations and TMs/TLs.