I have reread what you have stated, and then the Scripture came to mind about the Kingship. God promised David that the Scepter, or the Throne would not depart from Judah, until Messiah Comes. all the Kings up to Solomon were I believe chosen by God, after that I am not sure. kind of like the number of Priests that held the High Priest position, they were Anointed with oil and blood, and they lived long, until Solomon's temple was destroyed. after which the number were many, and they were only Mikvaed which is immersed in water! 7 times, and that was their Anointing, and they didn't live long. Also the people wanted a King to be like the other nations, and that rely displeased God, But He allowed it! After Solomon they also went through Kings, they didn't hold that position very long.
though the Lord brought Saul to Samuel to show Samuel who to anoint king I do believe the Lord told Samuel that Saul was the "people choice" if you will, and then after Saul did what was in his own judgement of what was good in the case of letting the Israelites have the live stock after they attacked a certain people, which was not according to what was written in the Torah and what the Lord reiterated when instructing Saul to do so.
after that, David was a chosen of God, and the Lord said as much. also I'm sure you remember that the Lord had Moses write down the info on the fact that one day they would have a king. but your correct on the count of one of the reason they would have a king like other nations, but again that is an observation on their part that kings that are men are obligated to protect the kingdom no matter the behavior of the people were as the Lord would leave Israel to its own demise until they turned to Him in repentance, of which a king that is a man like them, wouldn't require such.
I don't think the anointing is the key to why kings aren't permitted in the temple, but that doesn't mean that the anointing isn't the reason, it just doesn't seem to be the issue if you know what I mean.
and chosen is through the life of, when it comes to the way of the Lord on this. after Adam God saw man through Adam, after Noah God saw man through Noah who found "Grace" in God's sight. after that God chose Abraham, not randomly mind you there is ten gen's from Adam to Noah the ten Gen's from Noah to Abraham. hence Abraham was chosen since Adam. but to continue the Israelites are seen through the life of Abraham and the distinction is there, Paul said something to the order of gentiles through faith and Jew by Faith.
Rom 3:30 Seeing
it is one God, which shall justify the circumcision by faith, and uncircumcision through faith.
the Israelites were already believers it was just a matter of acknowledgement of Christ. anyway I digress.
but in the case of David again God chose David and through the life of David did God establish to coming Christ who is King and will return in such glory. also God sees us who are born again through His Son our Lord Jesus Christ and therefore is the same that He sees us through the Life of Christ.
the connection between the king and the King the priest and the Priest is obvious, the tabernacle had to be established first which requires priesthood. a priest has a relationship with his God or he is not a priest. kings establish and maintain kingdoms not tabernacles. and in the case of those born again what is the tabernacle of the Presence of God? hence where is the Kingdom to be established without that?