Usage of <y> (and <th> for that matter) in place of <þ> predates Caxton's press. An example of a scribe substituting <y> for <þ> in manuscript can be seen at [1], a legal document in an English chancery hand from c. 1445. The use of <th> was already very common in the Middle English period, with <þ> (or later <y>) retaining use only in sigla (common scribal abbreviations). See [2], a page of Chaucer's Canterbury Tales from the Ellesmere manuscript (c. 1410) for example.

In fact, many (most?) of the works printed by Caxton himself actually do feature a distinct <þ> glyph, though it is limited in use to sigla (it never appears without a superscript <e> for <þe>, <t> for <þat>, …). His first printed English work [3], printed c. 1473 while he was still stationed at Bruges in what is now Belgium, has a distinct <þ>; his later works printed at his own press in London do also, such as [4] (which is printed c. 1480 in the typeface characteristic of his press).

For some related discussion, see [5].

That said, I share your dismay that we no longer have a <þ> (or, honestly, an <ð> or a <ȝ>, though [ð] and [θ] are allophones and the <gh> once represented by <ȝ> is now often silent). I just wouldn't be so quick to blame Caxton.

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[1]: http://medievalwriting.50megs.com/scripts/examples/chancery6...

[2]: https://upload.wikimedia.org/wikipedia/commons/e/e0/Ellesmer...

[3]: http://lcweb2.loc.gov/cgi-bin/displayPhoto.pl?path=/service/...

[4]: https://upload.wikimedia.org/wikipedia/commons/b/bc/Brut_Chr...

[5]: https://news.ycombinator.com/item?id=15678583