The 'scent-path-or-gradient' doesn't apply here because it's too hot in the sahara for a scent to last long. additionally, there aren't that many ants out there to produce enough scent - they would get burned rather fast (this, for example, is the reason why those colonies only send the old ants to collect food). Furthermore, the region one ant scans for food is rather huge (the move really fast, otherwise they 'would burn their feet')... For more information on the ants google for Cataglyphis.

Say, for example, the ant is about 45m away from its base, and to replace it somewhere else, the ant will nevertheless run about 45m (and afterwards, if it can't find its colony, it will start with search-patterns). You won't be able to accomplish this with rhythm only.

If you would have read the article (I didn't read it, but I'm from uulm and am familiar with the corresponding research results) or the referenced research, you would know that all what you stated is obviously taken account for. For example your "they only have to remember a picture of all the puppies and notices the picture"-statement is wrong because of the following experiment: An ant is taken from its current location and moved to another location some meters away. So the ant has no way to tell where it is located at, but it still runs the straight way "home" (though ending not at the ant colony but somewhere else)...