[RPG] Does a Arcane Trickster’s Spell Thief return to the target on death

Yes, but only 1st-level spells

PurpleVermont has already argued why such a multiclass could only write down spells of a level they could cast as a Wizard (so actually up to 2nd level, if they take Wizard 3). I disagree with Vermont's argument that stolen spells could not be transcribed at all, however.

The rules don't clearly state that you have to have seen a spell written down to be able to transcribe it, that's a matter of interpretation--and I disagree with the interpretation:

Copying a spell into your spellbook involves reproducing the basic form of the spell, then deciphering the unique system of notation used by the wizard who wrote it. You must practice the spell until you understand the sounds or gestures required, then transcribe it into your spellbook using your own notation.

The fact that different wizards use different notation implies that the written form of the spell is not an inherent part of the magic, merely an aide-memoire. And the requirement to practice the spell until you understand its V/S requirements -- precisely the thing which the Arcane Trickster can instantly intuit -- strongly suggests that this is the most important part of "learning" a spell, and the rest is merely paperwork that your first-level Wizard training would absolutely prepare you for.

Although many tables allow some degree of tweaking in order to make a build match a concept, they usually involve some degree of horse trading, i.e. giving up something to get something else. Assassins get the Assassinate ability at third level. Arcane Tricksters get Mage Hand Legerdemain. Wizards get their own specialties. Unless he is willing to give up something from one of his other classes in order to justify the Legerdemain, you are certainly on firm ground to call him on this. Getting something for nothing is the definition of unbalanced.