Yes, the spirit can serve as a mount
It would need to be a pretty sad DM that would not allow this:
1. There is a long history of Dragons serving as mounts through all editions of D&D. Just look at Dragonlance. Even in 5th Edition we can find many published examples:

Mordenkainen's Tome of Foes, page 16 (above)
The Monster Manual entry about Githyanki calling them Red Dragon Riders
In Fizban's the Dragon Rime has this description: "This now-ancient white dragon was raised in captivity, bred as a mount for an ancient knightly order,
A table in Fizban's on page 114 says "An emerald dragon seeks heroes to participate in a dramatic reenactment of a legendary battle involving dragon riders"
In Rise of Tiamat we get for example on p. 48: Some survivors claim to have seen a cultist riding atop the dragon. Most of those refer to the dragon rider as human, like the other cultists. However, a few people report that the figure moved with the grace of an elf.
In Waterdeep: Dungeon of the Mad Mage, p. 217, we get: A male githyanki knight named Raajna and his mount, a female young red dragon named Smoakcant, are on duty here.
So, yes: dragons have "an appropriate anatomy to serve as a mount" in fifth edition, we have lots of evidence for that.
2. The draconic spirit has the physical form of a Large dragon, and is corporeal. It says so in the spell description:
You call forth a draconic spirit. It manifests in an unoccupied space that you can see within range. This corporeal form uses the Draconic Spirit stat block. (...) The creature resembles a dragon of the chosen family
And the stat block creature type line is: Large Dragon, Neutral, so it is large enough to serve as a mount for a medium or small creature.
The spirit has a Strength of 19, so it will be able to carry a rider with gear and barding weighing at most 570 lbs.
You can ride a dragon in 5e, and the Draconic Spirit has the physical form of a large dragon. Ergo, you can ride it.
It is another question how well trained you are for riding an exotic, flying mount, what it will take to not fall off if they make maneuvers, or what kind of saddle you need to get manufactured. But ride, you can.