Why did 'y' disappear as an internal vowel in English spelling?
The spelling change from 'y' in Middle English to 'i' in Modern English in such words as wife or time is actually a consequence of the phenomenon known as the Great Vowel Shift.
In wikipedia's chart you can follow the path for the sound now in time in the leftmost column. And the corresponding IPA steps are summarised as follows:
Middle English [iː] diphthongised to [ɪi], which was most likely followed by [əɪ] and finally Modern English [aɪ] (as in mice).
Old English
Going back to Old English, the most common spelling for wife and time would be:
For wife wīf and it would be pronounced something like "weef" /wiːf/1 (actually the bar over the 'i' is a modern typographical convention to distinguish long from short vowels as OE does not have this distinction in spelling).
For time tīma and it would be pronounced "teema" /ˈtiːma/1.
The letter 'y' in Old English does exist but it represents the sound /y/ (as in German 'ü' or French 'u'). See for instance lȳtel /ˈlyːtel/ => "little". Wīf insteadis is pronounced with a long i /i:/.
In Old English the spelling wyf would have been a spelling mistake - the correct form being wif.
Middle English
The upheaval triggered by the Norman Invasion, which eventually gave birth to Middle English was marked, among other things, by a change in the spelling conventions. The usage of the letter 'y' was generalised for all words with the sound /i/ or /i:/, thereby following the rules applied in medieval French.
Therefore, the spelling of /wi:f/ as wyf became the rule.
Late Middle English / Early Modern English
Indeed a close examination of the very quote included in the question suggests that the 'y' is pronounced 'ee' as in beauty and not 'eye' as in why. This is visible in some of the words I have highlighted.
I amongst other have indured a parlyament which contenwid by the space of xvii hole wekes wher we communyd of warre pease Stryffe contencyon debatte murmure grudge Riches poverte penurye trowth falshode Justyce equyte dicayte opprescyon *Magnanymyte* actyvyte foce attempraunce Treason murder Felonye consyli … [conciliation] and also how a commune welth myght be ediffyed and a[lso] contenewid within our Realme. Howbeyt in conclusyon we have d[one] as our predecessors have been wont to doo that ys to say, as well we myght and lefte wher we begann.
As you probably guessed many of the words above are verbatim French spellings. I've checked online for instance the words justyce, felonye and penurye.
Also, keep in mind that the GVS only affected long vowels, so that not all the words above spelled with the letter y are now pronounced with the sound /iː/.
The thing to notice is that, at the time of Thomas Cromwell the normal spelling of wife was wyf (Middle English) or wyfe (Early Modern English) and that it closely matched its pronunciation (/wɪif/).
Modern English
However, the Great Vowel shift was only starting and the next step after passing from OE wīf /wiːf/ to ME wyf /wɪif/ would be to pass to eModE /wəɪf/ and eventually to ModE /waɪf/ wife.
As the pronunciation shifted, so did the the spelling. The most common letter for the diphthong /aɪ/ being the letter 'i' 2the new spelling for /waɪf/ became our familiar wife.
Note 1 The spelling used for Old English is a new system at the time of king Ælfred. It is based on an extended Latin alphabet and it closely reflects the pronunciation of the time. Although there are inevitably some spelling variants, we are pretty sure of the pronunciation of such common words as wīf and tīma.
Note 2 Looking no further than the first person pronoun 'I'. Interestingly enough this can also be observed in the same quote if one considers the word contenwid (which is actually included twice but with slightly different spellings).
Let's look up the history of the letter 'Y'. Originally, it was used in Latin to write Greek loanwords. In the Greek language, there is a letter called the upsilon. The letter "U" was used by Romans at first to represent this letter(in Greek loanwords), as this letter was pronounced 'ü,' but when the pronunciation of this letter changed to /y/, the letter Y was used instead.
Old English took this letter and used it in it's words. However, in the First Grammatical Treatise, the letter was analysed to represent a "v" sitting atop an 'I' and the pronunciation thereof was now changed to 'VI'.
However, due to the Great Vowel Shift, the letter Y's pronunciation was changed to "WI", and by the time of the Middle English, Y had lots its roundedness and underwent synizesis assuming the same pronunciation as the letter 'I'.
Y now started to be used popularly, especially in the vicinity of ranging or 'minim' letters : m, n, and u.
So this explains why 'y' was used, but why did it not stay that way?
During the early Modern English Period and late Middle English Period, many of the words were respelt according to their Latin etymologies, and the spelling were then made official.
Thus, today, time is spelt 'time' and not 'tyme'