there are some problem here do you support the idea that because a Hindu can't eat cow meat let ban it for all
a jew can't enter dome of rock because he can't walk over the place of inner sanctum of Solomon temple and they are not allowed to walk over it , a Muslim have no such reservation.
your solution is equal to say because a jew cant work on Saturday let ban working on Saturday for every one .
and the rest of your suggestion is logistically not feasible (honestly from where you get the idea to rotate between mosque ,church and synagogue) , as i said a jew can't enter temple mount unless it become clear where is the exact place of inner sanctum . honestly a rabbi can explain it far better than me.
right now Jews and Muslim praying there as close as they are religiously allowed to go near temple mount.
by the way as far as I'm aware everybody can visit Al-aqsa but obviously not in praying time and please if you don't start another uprising don't come with armed escort or start preaching Zionism teaching there. about dome of rock the ban on non Muslim is the result of recent clashes with Palestinian and Israeli . if memory serve me correct non Muslim could enter it around 2000 and before that my guess when the situation calm down there they again can enter it.
you see such incidence is the reason non muslim barred from entering Dome of Rock.
https://www.eappi.org/en/blog/who-is-allowed-on-the-al-aqsa-mosque-compound
if these small extremist groups get checked every one would be a lot happier and many restriction will be removed.