Yes, a target can be chosen when the spell is unleashed.
This part is exactly the same as making a melee or ranged attack. On PH p. 194, the first step to making an attack is "Choose a target." Choosing a target is as much a part of making a ranged attack, as it is casting a targeted spell.
Yes, there are limitations to what targets are valid for certain spells, which don’t apply to weapons. But the mechanic is the same: take an action to ready your attack, a valid target comes into view, choose the target, resolve the attack. In either case, the action requires a target which may not be available when the action is readied.
Of course, there are differences between readying a spell and readying an attack with a weapon. These are delineated in the last paragraph of the "Ready" section (PH p. 193). (The spell must have a casting time of 1 action. The caster must end concentrating on another spell, if any. The caster may lose a spell slot, if concentration is broken.)
Nothing seems glossed over in the “readied spell” description: the restrictions are sketched out with some elaboration — but there’s nothing regarding targeting in these restrictions. Targeting is covered by the general rule on readied actions.
A spell’s requirements are accounted for when it is completed.
The requirements of a spell are generally accounted for at the completion of a spell being cast. For example, spell slots are expended when the spell is completed.
When you cast a spell with a casting time longer than a single action or reaction…if your concentration is broken, the spell fails but you don’t expect a spell slot or scroll.
PH p. 202, Longer Casting Times
(A readied action works a little differently than this, in that the spell slot is consumed if the spell is interrupted.)
Don’t get hung up on the one phrase Cast a Spell
The argument against this position pretty much hinges on the interpretation of the meaning of "casting a spell." But this term is used pretty loosely in the rules.
The specific mechanics are for various situations are typically laid out (for when spell slots are expended, or not, etc.) Pay a mind to the descriptive text about what happens when spell is cast in various situations. Levy those prices, but not more.
The requirements of spell casting are clearly laid out. Don’t assume there are more, hiding in between the lines.