Incorrect on a few accounts, brother.
Shirk was banned from the Ka'bah, by the actual ayat in the Quran. Some who did not convert actually did leave Makkah, most converted. Could you please provide the source of the above three conditions?
Every other non-Muslim region that was conquered thereafter was given the following choices before the assault,
1) Embrace Islam
2) Pay
jizya
3) Fight
After Makkah, the Muslims marched towards Ta'if, having learnt of their previous plans to flank the Muslims during the siege of Makkah. They met and defeated Hawazin (Ta'if's allies) at Hunayn. They then went on to besiege Ta'if. The siege failed and the Muslims returned. After a while, being surrounded by Muslim settlements Ta'if decided to negotiate. The negotiations ensued for a while until the people of Ta'if accepted the Prophet's (S.A.W) demands and converted. They were crying and wailing in the streets when their idols were being destroyed even though they had officially "converted". Mentioned this to show that all did not choose to embrace Islam and those who did, not all of them did it for the faith itself. It was after Tabouk that tribes from all across Arabia started sending their delegations to pledge allegiance to the Prophet (S.A.W) and that is how and when most of the peninsula became Muslim. Some still didn't, though.
Plenty of non-Muslims remained in the rest of the peninsula, entire towns even. Banu Taghlib was a Christian tribe who did not convert until long after the advent of Islam (They even had a false prophet). There are native Christians present in Kuwait and Bahrain to this day. Banu al-Harith is a Jewish tribe whose descendants are still present in the peninsula. Many more examples even closer to Medina who were left alone until Hazrat Umar's (RA) Khilafah when they were acted against for different reasons.
“No two religions”: Non-Muslims in the early Islamic Ḥijāz by Harry Munt is a good read. This idea about the peninsula was introduced later.
I mean why would Islam forbid non-Muslims from all of the Arab Peninsula anyway? The whole peninsula isn't holy land. What is the significance of the peninsula itself? Nothing, it's just a geographical feature. It had no unified history, politics, identity until the Muslims united it. It is Makkah and Medina, by proxy of being the Prophet's (S.A.W) resting place, that are religiously important. Otherwise, Jerusalem is far more significant to Islam than any other part of the peninsula.
There were 3 false prophets after the Prophet's (S.A.W) death. 1 woman (Sajjah who was a Christian) and 2 men (Musaylamah and Tulayha). Hazrat Abu Bakar (RA) fought wars against them not because they were non-Muslim or because of their false prophethood but because they refused to pay zakat as per their agreements with the Prophet (S.A.W) himself, i.e. treason. There was a massive debate on this amongst the Sahabah in Medina before the wars. Hazrat Umar (RA) advised against the war due to the politically volatile circumstances of the state after the Prophet's (S.A.W) death. This is all very well documented. People deliberately misquote this event to misguide us and further their own agendas against non-Muslims (Not you. Our mullahs).