Yes, the Moon Blade gets all the benefits of a Vorpal Sword
As you yourself quoted, the DMG states that when you acquire that rune, your moonblade functions as a vorpal sword. Vorpal sword does three things, a +3 to hit/dmg, ignore slashing resistance, cut off a head on roll of 20. That means so does your moonblade.
But isn't that too strong?
No, not really. First of all, a moonblade is a Legendary item. These things are meant to be rare, desirable and highly powerful compared to a regular magic item. Second, it's not at all easy to acquire the runes.
If the blade accepts you, you become attuned to it and a new rune appears on the blade. You remain attuned to the weapon until you die or
the weapon is destroyed. (DMG 217)
The blade only acquires a new rune when it gets a new master. So, unless your DM plans to continue the campaign through the lineage of your character, you're only ever getting one additional rune to what the sword comes with.
The other runes(at least 2) are decided by the DM.
A moonblade has one rune on its blade for each
master it has served (typically 1d6 + 1). The first rune
always grants a +1 bonus to attack and damage rolls
made with this magic weapon. Each rune beyond the
first grants the moon blade an additional property. The
DM chooses each property or determines it randomly on
the Moon Blade Properties table. (DMG 217)
So, right off the bat, you can see that the rules allow the DM to set the power level of any Moonblade that appears in their game. They can straight out choose the existing runes. But even if they do not, the rune table makes it difficult to roll up a completely broken sword.
Defender can only be acquired by 2/100 results.
Vorpal Blade can only be acquired by 1/100 results.
Elfshadow can only be acquired by 1/100 results.
In fact, rolls on 1-80 only give you a small boost of power, such as a +1 to the weapon, or a random minor property. The chances of rolling a truly broken weapon are extremely minuscule, and also require the DM to choose to roll the traits, rather than manually assigning them.
The Vorpal Blade rune seems powerful because it is. It is difficult to acquire by design, and almost impossible to be abused without DM's direct approval, since they have ultimate say over what runes already exist on your Moonblade.