Meaning of "I would there were..."?

What is the meaning of "I would there were", as in this quote from Shakespeare's "A Winter's Tale"?

I would there were no age between sixteen and three-and-twenty, or that youth would sleep out the rest;

From its context, I assume it means "I wish there were" but I find it difficult to conceive how the modern usage of "would" could ever have been used in this way.

Solution 1:

It simply means “I wish” or “O that”. Even today some people will still say “Would that it were so!” in an intentionally archaic manner.

See sense 36 of the OED entry on will. This is just an excerpt:

36. Similarly with const. as in 2: viz. with obj. clause, with vb. in past subj. (arch. except in would rather or sooner = ‘should prefer’), †rarely in pres. subj., or with acc. and inf. Hence (arch.) with ellipsis of 1st pers. pron. as an expression of longing = ‘I wish’, ‘O that’; also, by confusion with 37, in the form (I) would to God (or heaven).

• 1590 Shaks. Mids. N. v. i. 255, ― I am wearie of this Moone; would he would change.
• 1595 Shaks. John iii. iv. 48, ― I am not mad, I would to heauen I were.
• 1599 B. Jonson Cynthia’s Rev. To Rdr. A 4 b, ― I would thou hadst some Sugar Candyed, to sweeten thy Mouth.
• 1777 Miss M. Townshend in Jesse Selwyn & Contemp. (1844) III. 260 ― This news I picked up at Bet’s door. Would to God that we had peace!
• 1816 J. Wilson City of Plague ii. i, ― At a sad hour the sailor hath return’d; Would he were yet at sea!
• 1831 Scott Ct. Rob. xix, ― I would to God I had more.
• 1865 Whittier Kallundborg Church 48 ― Would I might die now in thy stead!
• 1882 Tennyson Charge of the Heavy Brigade Epil. 10-11, ― I would that wars should cease, I would the globe from end to end Might sow and reap in peace.

Solution 2:

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.