Spells do what they say they do.
From the description of Leomund's secret chest, we have the following:
You hide a chest, and all its contents, on the Ethereal Plane. You must touch the chest and the miniature replica that serves as a material component for the spell.
Emphasis mine. While duplicating the spell through casting wish may remove the need for the material components required to cast the spell, this does not alter any clauses of the spell's description (its effect). Note that the in the latter sentence, the singular is used: "...the replica that serves..." as opposed to "...the chest and the miniature replica that serve...", meaning that only the replica must be used as the material component of the spell. Either way, if there is no replica used in the casting of the spell, you are unable to touch it. While this on its own is confusing in how to handle, there is another part of the description which is clearer:
While the chest remains on the Ethereal Plane, you can use an action and touch the replica to recall the chest.
Emphasis mine. This means that even if the spell doesn't fail on account of not being able to touch the replica (and chest), if no replica is used in the casting, then the chest hidden by the spell cannot be recalled.
We would still be able to send some chest into the Ethereal Plane, though, as the indefinite article, 'a', is used, instead of the definite 'the'. This may serve as a means for the caster (or another creature) to enter the Ethereal Plane, or to easily dispose of some equipment.
One would have to make similar judgements for other spells which make reference to their material components, but hopefully this answer serves to present good principles with which to approach the problem.
Considering another interpretation...
The wording of wish is just vague enough to allow another possibility: though it's easy to assume that the 'requirements' which are removed are for casting the spell, that's not what the description necessarily states.
You don't need to meet any requirements in that spell, including costly components. The spell simply takes effect.
Emphasis mine. Under this interpretation, we have another set of outcomes. Considering again the spell Leomund's secret chest, then firstly, you no longer need to touch the chest and replica as stated in the first paragraph. This means that we unambiguously are able to cast the spell to send some chest into the Ethereal Plane without having a replica of it. However, we are still unable to recall it, as we are given the ability to recall it by "using an action and touching the replica"—which is not posed as a "requirement in the spell". Anything more liberal than this and we start to affect statments such as "Choose a creature within range."