What follows is just my guesses, as I haven't read any literature that brings up a special status for words ending in -ly.
-ly doesn't feel like a special case to me
I think that happy-tensing is a phonological phenomenon, not just a phonetic phenomenon. I can't tell whether the first video actually makes use of the phone [ɪ], but I definitely hear the phoneme /i/ and not /ɪ/ at the end of the word "really". And actually, I hear /ɪ/ and not /i/ in the first syllable. Both of these might be cases where what I hear is affected by my own accent, as I pronounce "really" as /rɪli/ and not as /rilɪ/ or /rɪlɪ/.
A phonological definition of happy-tensing would be that speakers with this tensing identify the vowel at the end of words like "happy" with the vowel in words like "freeze" rather than with the vowel in words like "fizz". That doesn't require that the vowels are articulated exactly the same. For comparison, some American English speakers have a tensing change where /ɪ/ has been replaced by the phoneme /i/ before a velar nasal, in the sense that they report that words like "king" have the same vowel sound as words like "keen" and "seam", but my understanding is that the actual phonetic realization of /i/ in this context may still be intermediate between [ɪ] and [i].
From an articulatory standpoint, your suggestion that [ɫ] might condition a more open realization of the "happy" vowel seems fairly plausible to me, but from a phonological standpoint, I don't think it's likely that speakers would identify words like "happy" as having the phoneme /i/ but words like "badly" as having the phoneme /ɪ/. And even from an articulatory standpoint, I would expect that low stress should allow the /i/ in words like "happy" to be realized with a more open quality than /i/ in stressed syllables, even though "happy" doesn't contain [ɫ]. So overall, I'm not convinced that [ɫ] is all that relevant to whether a vowel is subject to happy-tensing.
I think that it may instead simply be a case of the "happy" vowel displaying a relatively high amount of free variation in its phonetic closeness while still being phonemically identified as /i/ by speakers of most American English and Australian accents.
There should be a way to test this: if the quality of the preceding /l/ is actually an important factor, that leads to the prediction that you should not hear any instances of the happy vowel sounding like [ɪ] after other consonants from American and Australian speakers (except for maybe after /r/, which can be articulated with a similarly backed tongue position).
the /l/ in -ly is not morpheme-final
I think that you've misapplied the rule that you cited for the distribution of clear and dark l. Although I think it's likely that a number of speakers have [ɫ] in -ly, that is not an expected result of a rule that says to use [ɫ] "in morpheme-final positions before a vowel". That rule would apply to words like goalie and mailing, which can be divided morphologically as goal-ie and mail-ing. The /l/ in -ly is not at the end of a morpheme: it comes at the start of the morpheme -ly.
Speakers who have [ɫ] in words like badly or highly would have to be following a rule that says to use [ɫ] in more contexts than just morpheme-final positions. For example, they might be following a rule that darkens /l/ when it is not foot-initial, regardless of its position in a morpheme or syllable.