I can read the text two different ways. Your question seems like it's calling out to the possibility of the second one, but I definitely would not recommend using it, except if you're looking to mess with your players.
The logical interpretation : you benefit from the spell
The most logical way to read this is the following : you obtain the effects of Detect Thoughts and use its effect on the target creature. If you did not target a creature, the spell does not take effect and you take the psychic damage. Works as intended.
Good to note that, while it is probably intended that you can only use that ability on the target creature, nothing in the text prevents you from using that ability again on any other target, until the spell runs out.
The "cheesy" interpretation, or playing on the wording
The spell's description mentions the following :
When you cast the spell and as your action on each turn until the spell ends, you can focus your mind on any one creature that you can see within 30 feet of you.
There is no mention of targeting. You simply use the ability that is granted to you by the spell, just like you would look or speak to someone. This means that the only effect of the spell is to give the target of the spell this pseudo-mind-reading ability for the duration.
Normally the spell is limited in its possible targets by its range : self. However, the Wand of Wonder's description of the effect mentions it takes effect "on the target you chose". Specific beats general, and the targeting of this spell by this effect is more specific than the targeting of this spell in general, so the spell could target another creature.
Because of that, with that reading, the target of the spell could spend their action to use the pseudo-mind-reading ability, for the duration of the spell.
This obviously isn't the expected way to read and use that effect for the wand, but I believe that is the pseudo-loophole you wanted to talk about through this question.