No, you've misunderstood a few things about the smite spells
Casting Time: 1 bonus action
The most significant thing you got wrong was the casting time. All 7 of the smite spells have a casting time of 1 bonus action. This is significant because it means that you can cast the spell and take the Attack action on the same turn - presumably, you'd cast the spell first so that you can gain the immediate benefit right away, with no risk of losing concentration until after the immediate effect (assuming you hit).
Possible additional effect that lasts for the duration
The smite spells all cause the next weapon attack that hits to do additional damage (and all of them except banishing smite and branding smite specify "melee weapon attack") - but they all have an additional effect beyond damage that lasts for the duration of the spell, which is a significant part of why they all have a duration of "Concentration, up to 1 minute".
If they had no additional effect, there'd be literally no reason for a paladin to use the spell instead of expending the same spell slot to use the Divine Smite class feature; the extra damage of each of the smite spells is less than a corresponding use of Divine Smite, and only branding smite and searing smite can even be upcast to do more initial damage. (Though the paladin could always make use of both a smite spell and Divine Smite on the same attack just to do tons of damage at once.)
That said, the particular secondary benefit may vary between each smite spell:
- Banishing smite (PHB, p. 216) does 5d10 extra force damage to
the target, then banishes the target for the duration if it has 50 HP
- Blinding smite (PHB, p. 219) does 3d8 extra radiant damage to
the target; then, the target has to succeed on a Con save or be
blinded for the duration of the spell (it can repeat the save at the
end of each of its turns).
- Branding smite (PHB, p. 219) does 2d6 extra radiant damage; it
then makes the target visible if it was invisible and, for the
spell's duration, causes the target to shed dim light and prevents it
from turning invisible.
- Searing smite (PHB, p. 274) does 1d6 extra fire damage and
causes the target to ignite in flames, forcing it to make a Con save
at the start of each turn until the spell ends. On a failed save, it
takes 1d6 fire damage; however, the spell ends if the target succeeds
on the save, it or a nearby creature takes an action to put out the
flames, or some other effect douses the flames.
- Staggering smite (PHB, p. 278) does 4d6 extra psychic damage,
and forces the target to make a Wisdom save; on a failure, then until
the end of its next turn, it has disadvantage on attack rolls and
ability checks and it can't take reactions.
- Thunderous smite (PHB, p. 282) does 2d6 extra thunder damage
(and causes a thunderous sound that's audible up to 300 feet away);
the target then has to succeed on a Strength save or be pushed 10
feet away and knocked prone.
- Wrathful smite (PHB, p. 289) does 1d6 extra psychic damage,
then forces the target to make a Wisdom save or be frightened of you
for the duration (it can make a Wisdom check against your spell save
DC as an action to end the spell).
Staggering smite and thunderous smite are the only ones whose possible secondary effect doesn't last for the spell's duration; the former lasts until the end of the target's next turn, and the latter only has an instantaneous effect. (Neither one specifies the spell ends after that secondary effect is resolved, but there's no point concentrating on either of those two spells after that is done.)
A number of these secondary effects are quite significant. Banishing smite has a no-save banishment as long as the target's HP is low enough; taking an enemy totally out of the fight can be a pretty big deal. Blinding smite and wrathful smite impose a pretty significant status effect for the duration, though the save can be repeated each turn.
Staggering smite imposes a significant debuff, but only for one turn; removing the target's ability to take reactions may be the most useful part if strategically employed to prevent an ally's spell from being counterspelled or similar. Searing smite seems useful in doing continuing damage, but the repeated save in addition to the target's ability to remove it themselves or have an ally remove it (albeit at the cost of an action) weakens it somewhat.
Branding smite's secondary benefit is probably the most situationally useful; it's rarely needed, but when it is needed, it really shines (literally). If you know you're going to be fighting something that can go invisible, this spell can basically negate that entirely, and it's only a 2nd-level spell.
Thunderous smite seems the least useful at a glance; all it does is knock the target back and prone if it fails a save. Since it only applies to the next melee weapon attack that hits, this can't even be exploited to knock flying enemies out of the air unless you're somehow flying as well (in which case it's a great niche interaction). It also targets a saving throw that is relatively high among monsters, at least in the Monster Manual. Its saving grace is that it's only a 1st-level spell.
True strike works entirely differently
As Akixkisu's answer mentions, the true strike cantrip is substantially different. It has a casting time of 1 action, and also explicitly says its benefit applies "on your next turn" - preventing you from benefiting from it on the same turn you cast it, even if you have the 2nd-level fighter class feature Action Surge. It also doesn't impose any effect on the target, damage or otherwise; all it does is give you advantage on your first attack roll on your next turn.
Even setting aside its badly worded duration, the usefulness of the true strike cantrip is limited at best. At the very least, the smite spells do extra damage, and many also have a significantly useful secondary effect.