In Early Middle English will (of which would is the preterite and subjunctive form) had not yet declined into a mere modal; in both speech and writing it could have its original sense of “desire” or “want” (want at that time usually meant “lack”). Think for instance of the subtitle of Shakespeare’s Twelfth Night: or What you Will—that is, “whatever you want”.
That sense is still strong in the noun: free will, will and testament, the will of the people.
Would is particularly interesting; in a past context it might bear either the original sense of “want” or the modern modal sense, but in a present context it bears either a near-subjunctive or -irrealis sense — the “wish” you correctly identify here — or a conditional sense.
Here are some more uses of would in this sense from the first half of Winter’s Tale:
1,2 O would her name were Grace!
1,2 If you would seek us / We are yours i’the garden.
2,2 I would not be a stander-by to hear / My sovereign mistress clouded so ...
2,1 Would I knew the villain ...
2,3 So I would you did; then ‘twere past all doubt / You’ld call your children yours.
2,3 For life, I prize it / As I weigh grief, which I would spare ...
2,3 The bug which you would fright me with I seek.
2,3 ... for mine honour, / which I would free ...
3,3 I would you did but see how it chafes, how it rages ...
3,3 Would I had been by, to have helped the old man!
3,3 I would you had been by the ship side, to have helped her.