No, they don't need to know ...
I am not 100% certain that I fully understand the motivation of your question but since I just finished DM'ing that adventure for a group of D&D rookies a few months ago, I can tell you about my work and experience.
First, there's no need for the characters to know anything about a dragon. In fact, I didn't even tell my players anything about the title of the adventure. Partially, to prevent any metagaming, but also to build up real tension (see last section below about my greater plan).
At the beginning, all rumors might be just flavor for the (Dungeons & DRAGONs) story. But, it's also possible to (random) encounter the dragon on the first day (of course, that would be a 'You see a dragon swooping down in the distance'-encounter, only). And since even the youngling dragon is far too strong for the character group at level 1, it still might go as 'flavor' only. My group was especially 'unlucky'; they didn't actually see the dragon until they were at level 4, I think, when it tried to hunt the ox the group has borrowed (and it was an awesome experience for both players and characters as well).
The story slowly (!) builds up more evidence that a dragon is really causing trouble in that area and is causing the orcs (and others) to move into new locations. It's not important in the complete first half (or even 3/4?) of the adventure for the characters to know that there's a dragon involved in the story or even the center of the story.
The start: My character group was hired as (cheap) guards to protect a carriage to Phandalin, because it was told that in the past few weeks several merchants and travelers were ambushed by orcs (or kobolds?) in that area. There was no need to talk about any dragon.
You wrote:
The dragon that besets us is not the first to threaten this region.
That implies that the dragon is well known to the public by that point in the campaign, although it is not guaranteed that the players have heard about the dragon by then.
I do not agree to your interpretation, or rather: I adapted my interpretation to my greater storyline (see below; I decided to let dragons to be very rare at that time): That cited rumor is just from an old person, obviously trying to make himself important, (perhaps a dwarf, because of the age) that knows about that old dragon story. I still kept my players in the dark about the dragon: dragons are fabulous creatures from the past. Sure, they existed once, and maybe they are still somewhere, but not here at the Sword Coast: Most ordinary humans haven't seen one in their lifetime.
The last dragon in that area is just history - I don't remember if that was written somewhere, but I told my players that the last dragon that lived in that area was seen last several decades, maybe well over a century ago. That's the reason no one around knows anything specific about that old dragon or its lair. The sighting of a (new) dragon is somewhat fascinating - and frightening, of course.
The new dragon uses a different lair anyway than the old one; an old fortress, and even extended knowledge about the ruin's location is almost lost in the past. The exact position is not needed for the DM to know. I just told my player about their travel into the mountain region (after they learned about the fortress existence) and began to use a map again only, when the characters reached the spires (I assumed that there would be many, only the highest one is called Icespire Peak) and spotted the old fortress.
I hope this outline helps you. One last advice: Although this adventure is an introductory one for beginners it can quite easily drag on for a while, and even be stretched into a small campaign. But if you plan to do so; think about the next adventure or campaign. As I said, I planned for the Dragon Queen campaign, so I have used a lot of information from that campaign already in order to allow a very smooth transition. But in the end, I decided differently because of the way the story of the group was developing and what they were looking for. Instead, I extended the campaign by the 'Beyond the Dragon of Icespire Peak'-adventures, and that fits perfectly. Tiamat just has to wait :-)
