From the official rules (emphasis mine):
BANKER… Select as Banker a player who will also make a good
Auctioneer. A Banker who plays in the game must keep his/her
personal funds separate from those of the Bank. When more than five
persons play, the Banker may elect to act only as Banker and
As many people who play auctions in Monopoly know, starting a typical English-style auction at a $1 minimum bid and allowing increments of $1 each makes for painfully slow auctions, especially for higher value properties. However, the Auctioneer's primary purpose isn't to make auctions fast, profitable, or even interesting; he's there to sell property. Selling property moves the game along, and the faster the property gets into the hands of the players, the faster everyone can get to the interesting trading and development part of the game. There should never be any cases where any property remains on the board after someone lands on it.
However, the rules explicitly state that the Banker needs to be a good Auctioneer. Any Auctioneer who lets all of his auctions run forever for no good purpose doesn't really qualify as "good". Adequate, sure. Competent, maybe. But not good.
Regardless of the starting bid, any decent auction will result in the property selling at a reasonable market price; the only variable is how long the auction runs before this price is reached. If the opening bid is too high, nobody will bid until the Auctioneer lowers it. If the opening bid is so low that everybody's bidding, he can make larger increments to weed out the cheapskates. He has many tools at his disposal to figure out what this reasonable market price is, and exactly which tool is needed depends on the situation.
An Auctioneer who knows how to control an auction as efficiently as possible to sell property is a good Auctioneer. An Auctioneer who does nothing but lets his players control the auction is just a Banker.
As the opening question states, properties will rarely if ever sell at below the mortgage value, if for no other reason than because other players don't want their opponents to get free money. As such, starting an auction at or near the mortgage price is perfectly reasonable in most situations.
However, most situations are not all situations. There will be cases where nobody actually wants the property at mortgage value; it could be because all cash is tied up in developments, it could be because people are saving up for that last property in a colour group, it could just be because nobody likes utilities. By having no fixed minimum bid, the Auctioneer thus has the option to sell the property at a loss, just to sell the property. That is his job, after all. Sell property.
Breaking one of the tools in his toolbox makes for a less efficient Auctioneer, and we don't want that.
The chances of all players being unwilling to spend $1 on a property, which they can immediately mortgage for what is effectively a full refund, is negligible; if the price is right, any property should sell. Fixing a minimum bid in the rules would open up too many chances of a property going unsold, which would inevitably slow down the game each time someone lands on that square and the whole process starts up again.
Even a property that's immediately mortgaged can eventually be paid off and activated, or traded to other players. An unsold property is just useless.