Our friendly Hobgoblin's answer includes lots of valid interactions that might apply to the specific special abilities granted one subclass or another.
SeriousBri's comment points to a more general concern, though - since you need to spend your Action to attack to get the Flurry of Blows, allowing the Flurry of Blows to occur first could remove your opponent from the field. You would have thus spent your Attack action but have no one within your reach / range left to attack.
Removing said opponent is not limited to landing the killing blow, though. What if the creature you are attacking has a reaction-fueled ability that triggers in response to taking damage, and which then denies you your Attack? There are many possible examples of this, such as an archfey warlock's misty escape, or a contingency spell with a teleport, etc.
Even in the absence of a spell or class ability, any creature your monk faces, at any level, could have a readied action whose trigger is being attacked or taking damage. If that readied action removes them from your reach (such as simply moving away from you) you could be left having declared an Attack action but without someone to attack.
On the player exploit side, note that your monk might expect to do more damage with their main attack than with one of their flurry of blows. They might, for example, be wielding a magical staff two-handed. They thus might have a single, higher-damage attack and two lower-damage attacks. If these can occur in any order, it would be to their advantage to use the lower-damage attacks to make the higher-damage attack more likely to hit tellingly.
You have already suggested one such situation, whereby the flurry attacks are used to attempt knockdowns in order to give the more damaging attack advantage to hit against the now-prone opponent. But there are plenty of situations in which prior attacks can make subsequent attacks more likely to hit, or more likely for their damage to actually count.
Your monk could also use their flurry of blows, for example:
to whittle away at a caster's mirror images, saving their big damage main Attack for when they have the best chance to have the damage go through.
to accept disadvantage against a displacer beast, hoping that a hit would end its displacement ability and thereby making their higher-damage hit more likely (thus increasing expected damage).
to 'use up' the disadvantage imposed by vicious mockery on a lesser attack before they attacked with something more powerful.
to discover the location of an unseen and Hidden opponent by attacking potential locations, and attempting to save their more damaging attack until they have confirmed where the opponent is.
Allowing the attacks to be used in either order opens up a range of tactical considerations which the monk can exploit, but also exposes the monk to having a wasted attack if their opponent can respond to the flurry.