You have to read advertising claims about frequency coverage with a bit of a
jaundiced eye in many cases: I've seen antennas that claimed to "work" on a
band when only 1% of the power was actually radiated. While many of the
untuned vertical types can be tuned to a low SWR across a wide frequency
range, ground losses are often high on the lower bands (160m and 80m especially)
due to the short length, and the radiation pattern may not be ideal (though
probably still usable) on the bands where the antenna is longer than 5/8 wave.
In addition, tuning the antenna through a length of coax operating at a high
SWR will also increase losses.
Here is one analysis: http://vk1od.net/antenna/multibandunloadedvertical/index.htm
The details might not be quite the same as what you are considering, but the
general principles should apply.
Sometimes in the practical world we need to choose options that may not be
ideal in other conditions. If all you can manage is a vertical on a single ground
rod then do it. True, your signal won't be as strong as it may be with a good
set of radials, but even the worst antenna up in the air works better than the
best one still sitting in a box in the garage. You can try using several ground
rods around the base, tying them together with heavy wire. Or you can use
heavy wire for the radials and let the dogs dig it up. Or put down welded cattle
panels or reinforcing mesh for concrete - let the dogs try to dig through those.
I have some 1 1/4" structural steel tubing from a greenhouse that collapsed in
the snow that you could use for radials that probably wouldn't be bothered by
the dogs, either. (Magnetic materials such as steel aren't as good as copper
for radials or antennas, but sometimes you need the durability.)
If you are using a poor ground, then there may be some advantage to choosing
a trapped antenna such as the 18AVQ or 6BTV rather than one that requires a
tuner, even though it may not be advertised to operate on as many bands. That's
because, on the bands where it is intended to operate, the coax losses are likely
to be lower.