The vector for “culture” has always been trade throughout human history. The Internet is the best means to wage the war of modernity if you wish to wage a cultural “war”. I say, yes, go ahead and bring Islam into the modern world. You’ll find it naturally moderating.
I don’t understand your final paragraph at all.
I agree with you on the fact that we are on the 21th century !
Great news .
I think that you have a liberal, ie wrong , understanding of the problem and a wrong understanding of what is culture .
Modernity is not a synonym of goodness and truth and progress in every fields of human activities . Or you missed some part of history or you have a fantasmatic rosy vision of modernity which is a kind of (fake)religion.
J.Christ lived 2000 years ago and Aristotles 2350 years ago... It’s absolutly not a question of age and time.
Islam is wrong and bad and always will be but we can influence some muslim people by culture ie science, philosophy,art and christian teachings.
Culture need individual learning and hard and patient work.
Internet is only a tool ! Depends on how and what for one is using it