With your houserule, the DPR is huge

Natural Weapons are not Unarmed Strikes, making them so is a very unofficial, very unbalanced houserule.

Level selection

The best damage available in beast form is 24(4d8+6, Triceratops Gore) at +9 to hit, but you need 15 levels of Druid for it. To get the Extra Attack, you have to have 5 levels of Monk.
This is all of your 20 levels.

Starting Ability scores

for Human Variant with the Resilient(Con) for Concentration:
Str: 8
Dex: 16
Con: 14
Int: 10
Wis: 16
Cha: 10

ASI Monk 4: Sentinel, for out of turn DPR
ASI Druid 4: +2 Wis, for AC
ASI Druid 8: +2 Dex, for Flurry of Blows
ASI Druid 12: +2 Dex

Calculation

Assume AC 20 for the enemy, it is not unreasonable on level 20.

Official DPR:
+9 to hit, 2 x 24 (Gore) = 0.5 x 2 x 24 = 24
+11 to hit, 2 x (3.5+5) (Flurry of Blows) = 0.6 x 2 x 3.5 = 10.2
Alltoghether it is 34.2 vs AC20.
This is a very sudden jump however, at level 19 you either do not have Triceratops, or Extra Attack. Without Extra Attack you do 12 + 10.2 DPR, below a 17th level Monk.

Houserule DPR:
+9 to hit, 2 x 24 (Gore) = 0.5 x 2 x 24 = 24
+9 to hit, 2 x 24 (Gore as FoB) = 0.5 x 2 x 24 = 24
Alltoghether it is 48 vs AC20.
Extra Attack makes a much smaller difference now, as the FoB does the same damage as attacks done with your action. This is very unbalanced.

Compare it to 17 levels of Monk:
+11 to hit, 2 x (1d10+5) (basic attack) = 0.6 x 2 x 10.5 = 12.6
+11 to hit, 2 x (1d10+5) (Flurry of Blows) = 0.6 x 2 x 10.5 = 12.6
This all remains the same in the next 3 levels; 25.2 vs AC20.

Extra Attack does not mean you take the Attack action multiple times

The Attack action

This is simply an action available to anybody and it allows the Monk to make one attack.

Extra Attack

Beginning at 5th level, you can attack twice, instead of once, whenever you take the Attack action on your turn.

This feature requires you to take the Attack action to receive its benefits and it causes your Attack action to involve making two attacks instead of the usual one attack. This feature does not allow you to take the Attack action twice, it simply modifiers how many attacks you make when taking the Attack action.

Arms of the Astral Self

Immediately after you use the Attack action with your astral arms on your turn, you can make one extra attack with your astral arms as a bonus action. The number of extra attacks increases when you reach certain levels in this class, increasing to two at 11th level and three at 17th level.

In order to benefit from this feature you must take the Attack action and at least one of the attacks made during the Attack action must be made using your astral arms. Using this feature requires you to use your bonus action and grants you one additional attack, at level 11 it grants you two attacks, and at level 17 it grants you three attacks. It does not matter how many other attacks the Monk made, this feature scales independently of all the rest.

Astral Barrage:

Whenever you use the Extra Attack feature to attack twice, you can instead attack three times using your astral arms.

This feature just scales up Extra Attack so your Attack action now involves three attacks and not two.

Thus we have all the data we need:

• 3rd level: You can make 1 attack with the Attack action and 1 using Arms of the Astral Self

• 5th level: You can make 2 attacks with the Attack action (because of Extra Attack) and 1 using Arms of the Astral Self

• 11th level: You can make 2 attacks with the Attack action (because of Extra Attack) and 2 using Arms of the Astral Self (because of how it scales)

• 17th level: You can make 3 attacks with the Attack action (because of Extra Attack and Astral Barrage) and 3 using Arms of the Astral Self (because of how it scales)

For some further related reading on the differences between an attack and the Attack action there is the following: What does upper-case-A-Attack action vs. lower-case-a-attack mean?