People probably do this strategy but they probably don't realize how ineffective and inefficient it is.
You really don't want to wait until a party member goes down, especially during certain initiative circumstances.
For example, if it goes monster, fighter, cleric and the monster knocks the fighter to 0, the fighter lost his entire turn and the cleric can spend his action to cast a cantrip or make the attack action at best. This is horrible action economy and the monster can easily knock the fighter out again next turn to reduce the party's action economy further.
If its monster, cleric, fighter and the monster takes out the fighter, its much more tolerable action economy but its still a less efficient than if the cleric just healed the fighter to close-to-full so subsequent rounds aren't spent casting cantrips and healing and spent putting better pressure on the monster.
I disagree.
You mention losing the fighter's turn.
If the monster is still threatening the location of the fighter, a cleric taking a turn to heal means losing the cleric's damage output (effectively losing a turn) while also failing to negate the threat. This means a possible net loss of 2 turns, rather than the fighter's 1 turn.
For similar reasons (if forced into a choice of one or the other,) modern combat troops are taught to negate/eliminate a threat rather than having a medic (for an example) run out to start triage on someone wounded by an enemy.
Dead enemies cease to have combat ability.
Edit: Additionally, there are other things to consider in the context of D&D.
Standing up a PC while they're still in danger means that they again become a target. Now, as a target close to 0, the PC is in danger of insta-death rules if hit and driven into the negatives.
Also, being that a healed PC becomes stabilized, the downside to waiting to heal is often less than the downside of the cleric spending several turns healing the ally.
Hypothetically, the DM could start to target the downed PC, but that still leads to an overall net benefit to the PCs because it means the enemy is effectively wasting a turn to hurt someone who can pop right back up after healing -meaning the enemy has 0 damage output against the rest of the party, while taking a round of damage from the party.
There may be cases in which standing up the downed PC is tactically better (such as healing the paladin so they can get up and do radiant damage to a foe weak to it). In those cases, healing the PC is the better option. However, in a vacuum (and absent specific circumstances which would indicate otherwise,) I would argue that it's better action economy and more tactically sound to mitigate the threat before turning to healing.