Mine stays outside up against the house and has always been fine. Of course I can’t remember if I turned of the propane or not but it will be on the patio when we return.

Good, I feel bad for all those poor little moths flying around without their balls.

since 1995 when we first purchased a second home in FL, I have always brought the grille into the garage when we closed up the house for the summer

clean the thing and you'll be fine

That is illegal here.

Yup. Mothballs work fine for all manner of pests

Storing gas grill.
 

For the past 14 years, I have taken off the tank and stored in garage, and have put grill in glassed in Lani. If you have cover, store in garage. Obviously clean grill first.

The standard recommendation is that propane (LPG) tanks should always be stored outdoors, in well-ventilated areas. Storing propane tanks in garages or unventilated sheds is discouraged because if a valve isn’t fully closed, or if there is a leak vapors could escape and concentrate indoors. A garage has many sources for ignition such as light switches, garage door openers, refrigerators, etc. which could cause an explosion of accumulated propane vapors.
A flat, level outdoor area that is out of direct sunlight is the ideal location. Be mindful of other flammable materials on your property and take care to store propane at least 10 feet away from those items. A fire close to a propane tank could cause it release propane and intensify the fire
Just because you never had a problem with storing your propane tank in your garage does not mean it is recommended. Go to your fire department or lpg refiller such as Amerigas or Blue Rhino for their recommendation and you will find that their recommendation will be consistent with the above.
Why take a chance on an LPG leak destroying your home. As advised by many above, take your LPG tank out of the grill tie it down outdoors and be safe.

It is illegal in Florida to use mothballs to keep away wildlife.

I used to store a full one in my garage with nary a concern.