Before Edit:
All five logos face one issue and that is they cannot be used for what I would classify as your target audience. Per your information
Rooftop patio with yard games such as corn hole, washers, basketball
hoop, darts
1st Logo

Outside on a rooftop this logo would be ideal for big large print. It could be displayed as a one color with no issues. Although for large print areas like a wall or canvas mural a horizontal logo would fit better. This could work if it was on a door that led to the patio or outside. It could work on a corn hole board because it is vertical. This wouldn't work on a basketball hoop because a horizontal logo would be ideal. I doubt the engraving on darts would be possibly with the Xs. If you were to get this embroidered on a polo shirt most companies would not be able to stitch something so small.
2nd Logo

Hard to read from long distances, reduce the logo to ten percent and all I can tell is a jug. It could work on a mural and patio awning but still the & and Co are out of place. The Hooch should be larger but then you risk of distortion within the jug. If this were to be done in etchmark on a large window it could work but for this logo to really work it would have to be large. This wouldnt work on darts, washers, or apparel. Only way it would work on apparel is if it was a black shirt and the logo took up the entire width of the allowed screen when you went to screen print.
3rd Logo

I think this is my favorite but as stated with all when you get to a small size the jug in the circle will not come through. I would work on omitting this. I would also look into just one solid color. and it would look good on a rooftop patio, awning with a black background and a logo in all white. Again, look at this at 10% and the jug at the top and ATX should be removed. This logo could also work if you were to use it to print labels on beer bottles.
4th Logo

Remove Prime Times it collides with logo. Possibly consider lowering the & Co and remove the condensed type. I would possibly look into testing adding space and stretching it.
5th Logo

Not really any different than the 1st other than a color change.
Overall you should test and develop the logo as one solid color first then expand the colors. You need to consult with the owner on all forms of medium they end to use. Look at some logo mockups or do a search for logo mockups. Dont be afraid to test against what they intend to use the logo for. Make sure to right down the pros and cons of the logo and why it may not work on X this or X that. There are some you can buy that you can test against like on glass bottles, walls, awnings, building glass, etc. etc.
If it was me I would go with the 3rd logo and remove the jug at the top and atx at the bottom. Possibly test and lower the $ co to the bottom of the turn on the H.
After edit:
What other objects are associated with moonshine, Hooch besides the
obvious bottle?
The distillery, old school vehicles, mob bosses, etc. etc. Do a research on moonshine and the prohibition era that influenced moonshine. You could even scope it to a region. Did the region your client is in have a roll in the shine age?
Do you think the "H" With a handle on it is strong enough for a stand
alone icon?
That could work but you should ask are you just trying to deploy the symbol alone or have this co-exist with a logo is a question you should ask yourself.

What can be done concept "E" to make it a stronger mark? Does it read
like an H?
Concept E, to me would be very complicated to use in smaller size and if you want to use it try to simply it by possibly removing the XXX and testing.
To me a simple logo like this I would test against different mediums:

After suggestion of lowering the "& CO":
