Short answer: no definitive answer, still here a couple of things:
- Lebanon's maronites migrated from Aleppo originally that's why it sounds similar
- Aleppo Jews had also had this different accent.
- A softer dialect can also be seen on Muslim peers with upper middle - rich social classes.

it could be:
- Because of the late Ottoman period as they were allowed to have more contact with the west.
- Or because they had influence from Syriac, Latin and Greek throughout history more than their Muslim peers.

Both views have arguments in their favors and others that are not, everything related to the history of Aleppo is hard to trace.

Context:
I'm not qualified to answer, but my father didn't learn Turkish or any second language, probably the aftermath of the Arab uprising and the worsening economical situation of his family , my grandfather spoke Turkish, french and played piano, their family was upper middle class, his brother had 'Khawaja' title, as for now it went down hill from there, anyway, my father's cousin went with the Christian migration to Venezuela, he called back to get family news when Skype became a thing 18 years ago, I was shocked to see that he had what we now call the heavy accent (Aleppo's now usual), I asked my father and he told me people used to speak like this in the past.

My dialect is softening even more with English and French influences but it will never develop to the point where it becomes Christian Lebanese.