How is anyone convinced of anything except by personal experience?
How does anyone come to Christ except by having an inner conviction of the truth of the gospel message?
How did Peter learn the Jesus was the Christ, the Son of the Living God? : “Flesh and blood hath not revealed it to you, but my Father which is in Heaven”
The Bible is replete with examples of prophets and apostles having personal experiences in which God transmitted truth to them — which they then went on to teach to others.
Personal revelations from Scripture, OTOH, are different. The person can point to Scripture, say "because Scripture says this and this, and this, I now believe this." There is a definite line a reasoning there. Others may not agree with the reasoning, but it's there, and can be identified.
Saying that you have had a vision, or a feeling, or something else, that you cannot explain, nor point to is not evidence, nor is it Truth. It must be backed up by external sources.
Otherwise, any ol' Joe can come along and create his own religion.
The Bible is replete with examples of prophets and apostles having personal experiences in which God transmitted truth to them — which they then went on to teach to others.
You are familiar with the passage in the Bible where Paul commends the Bereans. The above two questions are phrased in such a way as to support another way unlike the Berean way, a way that is not backed up by the Bible, and that is by design. Well, not backed up by the Bible in context, because we have witnessed the twisting of single verses and portions of verses in order to support a non-Christian assertion.
The Bereans made it their personal experience to study the scriptures to find the passages which would support or refute Paul's teaching their 'personal experience'. The Bereans didn't rely on mere feelings. Paul admonished Timothy to study the scriptures, to show himself, rightly dividing the scriptures. The foolish Galatians used feelings in the moment and were led astray soon after Paul left their community.
Let's return to Doctor Luke's scene as recorded in Acts; Luke was an eyewitness to these events about which he wrote:
Acts 16: 16-18 And it came to pass, as we went to prayer, a certain damsel possessed with a spirit of divination met us, which brought her masters much gain by soothsaying: The same followed Paul and us, and cried, saying, These men are the servants of the most high God, which shew unto us the way of salvation. And this did she many days. But Paul, being grieved, turned and said to the spirit, I command thee in the name of Jesus Christ to come out of her. And he came out the same hour.
This time when we look at that passage, let's find a few more morsels of data. Recognize that this girl with a spirit of divination was offering that 'These men are the servants of the most high God, which shew unto us the way of salvation'.
Your questions were: "Why would any demon want a person to believe in the reality of Jesus Christ? Would a demon inspire a person to repent of their sins and follow Jesus?"
Look at the scene, carefully. This demon possessed girl was doing exactly what your questions imply was not possible! BY telling people that Paul and his small band were servants of God, she is vouching for their position of truth deliverers. By her past use in earning certain men money over her divination, she is lending credulity and thus some people would be --even if only in a small way-- more inclined to listen and follow the teachings from Paul and the small group of evangelists. The girl specifies that these men, these servants of the most High God are showing the way to salvation!
So in answer to your specific questions, yes, this demon inspired girl was doing what your questions try to assert would not be done by a demon! Her speaking out the way she did is equivalent to Mormons' claims of 'a testimony', yet we have the Bible explanation that she was not inspired by the Holy Spirit else Paul would not have been grieved and sent the demon out of her!
But let's go a bit deeper into the statement by the demon possessed girl. Whom is making the assertion regarding Paul and the small band of Christian evangelists? If it were the girl speaking, I doubt Paul would have been so bothered as to after 'many days' turn and send the demon out of her. The demon was speaking through the girl!
And there at least two ther things in the telling, things very subtle Luke records: "These men are the servants of the most high God, which shew unto us the way of salvation." What is inferred by 'most high God'? Why that there is more than one god! And I'm sure Mormons want desperately to believe that, so would Mormons ignore that the girl spoke by inspiration of a demon, not by inspiration of the Holy Spirit?
You further offered: "How did Peter learn the Jesus was the Christ, the Son of the Living God? : “Flesh and blood hath not revealed it to you, but my Father which is in Heaven” The Bible is replete with examples of prophets and apostles having personal experiences in which God transmitted truth to them — which they then went on to teach to others."
Find one, just one passage in the Bible where something revealed to the Apostles was not or is not backed up by scripture? You can't. The Bereans tried and were commended by Paul since he knew Whom was author of their pursuit.
I can point to a myriad of things taught by Momronism's founding false prophet and subsequent leadership of Mormonism which contradict the Bible ... CONTRADICT! Is the god of Mormonism the author of confusion? Yes, absolutely. The god of Mormonism is the same inspiration which drove the girl in the scene in acts to speak out on whom these men were, and she repeated her proclamation for many days.
Knowing human nature, I have no doubt that she turned some away from listening to Paul and the other Christians because of the nuisance of her incessant shouting it out following them around. Some people want to ask their questions in a quiet, peaceful setting, not open to public scrutiny. The demon possessed girl was making a nuisance of herself and making the band of evangelists seem like a typical traveling circus of the day. A more current example are the private freepmails some of us get, asking for clarifications on scripture and the Gospel.
And just when the focus might have become the efficacy of what these evangelists were offering, the men who lost revenue due to the demon being sent from the girl diverted attention onto the loss of their revenues! Oddly, we see that exact tactic repeated on this thread when the focus is falling upon the lies and heresies exposed in Mormonism!
Jesus IS truth. Now that he has come and abolished the priesthood, we need nobody to tell us what is true except God's Word, The Bible.
Yes, I believe in personal revelation. And it's just that - PERSONAL. God may put it in your heart to do or not do something. That's for YOU! I know Christians who feel called by God to not drink for example, but that's just for them.
If he wants to tell me something, he will tell me, not tell you to tell me.
Get it? You may of course see me wandering away and point me back to the bible and prayer and try to help guide me.
But God is not going to tell you, or any other man, that everyone must now wear this, or eat that, or go here. Sorry. Jesus is my priest, not any man. I have a direct line.