Why use Clairvoyance when Find Familiar is better in every way (except range)

It's just a temporary inconvenience, except in a singular case.

Basically, all that a familiar's death means is that you'll have to recast Find Familiar at your earliest convenience (so it means the loss of a spell slot for a day if you have a time pressure and can't take the extra 10 minutes to cast it as a ritual). The spell/ritual does cost 10gp, but that's a pretty trivial amount for an adventurer. It also has a casting time of 1 hr so your party probably needs a short rest while you do it rather than simply doing it between encounters.

The only case where having a familiar come to harm is a permanent issue is in the rare event that you decided to retrain the spell (or as a wizard, have lost the spell book that contains it). These are the only times when losing a familiar has long term ramifications.

As to whether or not the newfound familiar obtained when you recast the spell is the same being reincarnated/resurrected, or a different being entirely, this is up to you and the flavor you choose for your familiar.

Yes, the familiar can use the effect of dragon's breath

The Find Familiar spell description says "A familiar can’t attack, but it can take other actions as normal". The term "Attack" means you're making an attack roll. You don't need to attack in order to use the effect of the Dragon's Breath spell (XGtE, p. 154), because there is no attack roll involved, but a saving throw instead:

You touch one willing creature and imbue it with the power to spew magical energy from its mouth [...] the creature can use an action to exhale energy of the chosen type in a 15-foot cone. Each creature in that area must make a Dexterity saving throw, [...]

Per the rules on making an attack, for something to be considered "an attack" in 5e, it must involve an attack roll or otherwise must be called "an attack" explicitly in the rules (as with grappling and shoving):

If there's ever any question whether something you're doing counts as an attack, the rule is simple: if you're making an attack roll, you're making an attack
(Player's Handbook, page 194)

you can use the Attack action to make a special melee attack, a grapple
(Player's Handbook, page 195)

"An attack involves an attack roll or doing something that the rules call an attack, like grappling or shoving"
(J. Crawford's comment)

See also: What counts as an attack?

Since the familiar isn't making an attack, it can use the effect of the Dragon's Breath spell.

Jeremy Crawford confirms this

As @V2Blast has pointed out, an unofficial tweet from rules designer Jeremy Crawford supports this answer:

Dragon's breath is cast on a willing creature. Your familiar is a creature that can be willing. Find familiar prevents a familiar from attacking, but dragon's breath involves no attack.