Why is the pronunciation of French loanwords with the ending é botched?

Take these French words that exist as well in English:

• résumé
• protégé
• sauté
• exposé
• café

The French pronunciation for the é is simply /e/, which exists in English.

So why is the widely accepted English pronunciation /eɪ/, rather than /e/?

For many speakers, /ɛ/ is not particularly close phonetically to [e]

We can transcribe the vowel phoneme in the word "bed" with /e/, but that doesn't mean it's phonetically identical to the vowel in French "café" etc., any more than the English consonant sound transcribed /t/ is identical to the French consonant sound transcribed /t/. Many English speakers pronounce the vowel in "bed" with a quality closer to [ɛ].

/ɪ/ (in stressed syllables, at least) is not thought of as an "e sound"

You're right that the English vowel phoneme found in words like "tip" and "bid" has a quality near the IPA cardinal vowel [e] for most speakers, but most English speakers don't think of /ɪ/ in stressed syllables as being an "e sound" so they're unlikely to use this vowel in a word where the stressed vowel is spelled with "e" or "é". For comparison, many American English speakers realize the medial consonant in words like writer and letter as something like [ɾ], but few of these speakers use this sound in loanwords as an approximation of the /ɾ/ or /r~ɾ/ sound of foreign languages.

There are also phonotactic explanations for why English speakers don't use /ɛ/ or /ɪ/ in the words that you list.

Phonotactic restrictions on /ɛ/ and /ɪ/ in word-final position

American English speakers tend to put the primary stress on the last syllable in loanwords from French, and stressed word-final syllables ending in /ɛ/ or /ɪ/ do not occur in native English vocabulary.

British English speakers are more inclined to place the stress earlier in loanwords from French, but unstressed word-final syllables ending in /ɛ/ also do not occur in native English vocabulary, and unstressed word-final syllables ending in /ɪ/ are only found (in words like happy, valley, taxi) in particular accents of English that I think have become fairly uncommon.

I think it's actually not particularly difficult for an English speaker to pronounce these sounds in this context (e.g., the slang word "meh" is pronounced /mɛ/) but it's not something that English speakers will tend to do without explicit effort.

In contrast, stressed word-final syllables ending in /eɪ/ do occur in native English vocabulary; e.g. in away, today and in a number of monosyllabic words like way, say, may, hay, gray, lay.

It's a bit less clear whether unstressed word-final syllables ending in /eɪ/ exist in native English vocabulary, but many speakers have /eɪ/ in the weekday names Monday, Tuesday, etc. (these could be considered to have secondary stress on the last syllable). Other speakers have a reduced vowel /i/ or /ɪ/ in this context; I give some further examples of the reduction of unstressed /eɪ/ to the "happy" vowel in my answer to Which English words feature reduction of diphthongs like /eɪ/ to /i/?

/eɪ/ is the established sound, and sounds "French" to English speakers

The English phoneme /eɪ/ has become established as the conventional equivalent to French /e/ (as well as word-final French /ɛ/, as in "ballet", and in some cases even word-medial French /ɛ(ː)/, as in "crêpe" and "fête"), and has furthermore become established as the vowel sound used in each of the particular words that you mention, so that's what people use.

Two words with unexpected pronunciation variants provide evidence that the use of /eɪ/ in English is not particularly closely related to the way a word sounds in French. The last syllable of the word word repartee, from French repartie, is fairly often pronounced /ˈteɪ/ in American English. Likewise, the last syllable of the word lingerie is often pronounced /ˈreɪ/ in American English. As far as I know, these pronunciations can't really be explained as any kind of attempt to approximate the actual pronunciations of the original French words; rather, they indicate that the use of /eɪ/ in the final syllable of words from French is an established convention in English (that might be vaguely related to what English speakers think French sounds like, but is not really related to what French actually sounds like).

Let's consider a comparable example in French. Why is the English word brunch pronounced /bʁœ̃ʃ/ with a nasal vowel, when there are lots of words pronounced with /n/ in French? Because whenever /n/ comes after a vowel and before a consonant, it is absorbed into the vowel, making it nasal.

The case of words ending with /e/ in English is similar. In English, /e/ and /eɪ/ are allophones, meaning they represent the same underlying phoneme, and /eɪ/ is always used at the end of the word. So the pronunciation /rezume/ is impossible in English, just like the pronunciation /brynʃ/ is impossible in French.

In French, a /j/ at the end of a word is phonemic; for example, vais (/vɛ/) and veille (/vɛj/) are two different words where the only difference is the /j/ at the end of veille. So when English speakers say sauté, French speakers hear sauteille, which would be a completely different word in French (if it existed).