Word to the wise about the Douay-Rhiems (DR) and the Wycliffe Bible (WB)*, they both used the Vulgate as their primary manuscript.** That’s why they are so similar, consider (in chronological order):
Vulgate: Omnis Scriptura divinitus inspirata utilis est ad docendum, ad arguendum, ad corripiendum, ad erudiendum in justitia // All scripture, inspired of God, is useful for teaching, for reproof, for correction, for training in righteousness. (my rough translation for comparative purposes)
WB (Early Version): Forsoth al ʽscripture of God ynspyrid is profitable to teche, to arguwe, ʽor proue, to reproue, for to lerne in riȝtwysnesse, that the man of God be perfyt, lerud ʽto al good werk.
WB (Later Version): For al scripture inspirid of God is profitable to teche, to repreue, to chastice, to lerne in riȝtwisnes, that the man of God be parfit, lerud to al good werk.
DR: All scripture, inspired of God, is profitable to teach, to reprove, to correct, to instruct in justice:
So here we have two/three English translations, none of which are translated from the actual Greek text 2 Tim 3:16. I would hardly say they amount to evidence of a mistranslation of the verse in question since they are translations of a translation. If anything, it is more likely they are reproducing an error in Vulgate than correcting modern scholarship.
* I include here the WB in the Old English, most websites that provide the Wycliffe Bible text online (like Bible Study Tools for example) use a later version which has the Old English deciphered. Bible Study Tools page on WB says:
This update of the Wycliffe Bible mostly follows the “Later Version”, but with irregular spelling deciphered, verb forms comprehended and made consistent, and numerous grammatical variations standardized.
** For those who may not be aware the Vulgate is the Latin translation of the Old and New Testament.