There are very specific rules about where translators have to be located if the AM station applies for it. You can lease translators. A more common scenario is to lease an HD FM channel and rebroadcast it on a translator. The HD signal can be the same programming as your AM station. And you can own the translator. This might sound confusing but a translator you get because you are an AM station has certain rules. The owner of the AM station is not restricted from owning 1,000 translators if they desire but the only way they can rebroadcast their AM station is to abide with the certain rules that apply. A loophole is you, as an AM owner, can utilize an HD FM channel and then rebroadcast that HD channel with a translator or two. That HD channel can be the same programming as your AM station. In commercial radio translators are generally restricted to the normal coverage area of the host station, which would be the HD channel in this scenario.
Yes, translators are like gold in many places because the FM dial is so crowded only a few eligible stations manage to get a translator. Looking at the coverage maps of some translators, they cover very few people so they are sort of useless but that's not the real value in the eyes of an owner, It's a translator that might be able to move to a better spot at some point. To move a translator, you must have one, and the FCC is not allowing new translators now.
As to if an AM could get a bunch of translators, I suppose you could if you meet the geographic rules and purchase translators after you are granted the one for your AM station. I'm thinking a 4 leaf clover styled concept where each leaf represents a translator coverage area with the stalk being the AM station site. In that respect, if it was at all possible, you could acquire three translators to go with your one you were granted, move as needed (if possible) to create the 4 leaf clover. The key is you need to stay within the geographic requirements for an AM translator. (For example, you don't see an AM translator for a San Antonio AM in Dallas because of the geographic restrictions)