Why do they expect Mormonism to be provable by secular tools when Christianity itself isn't? Bias.
If God wanted us to believe because of proof, wouldn’t He have left a very different set of scriptures? No; He wouldn't have left scriptures.
How do you respond when someone says they no longer believe because of historical problems? I attempt a root cause analysis to arrive at the the number one reason.
Have you ever felt the Spirit confirm something that your mind didn’t fully understand? Yes; that is a testimony, and we grow into it until we comprehend even God.
Why do you think God designed faith to require the Spirit instead of allowing clear, universal proof? A minor point, but faith does not require the Spirit. It requires the light of Christ which eventually leads to pondering and praying and then a confirmation by the Spirit, which is clear, universal proof to the one receiving it. But unless this spiritual knowledge grows, it will at some point lack clarity, universality and other attributes of good proof (e.g., relevance, sufficiency, applicability upon scrutiny).
I think the reason it works this way is that the jump from what can be done with the light of Christ to what can be done with a covenant relationship with Him requires His holy priesthood and grace, which are things we cannot properly, rightly or effectively use on our own (we would screw it up, just like we do secularly and "sectarianly" with the light of Christ!). We use the light light of Christ all the time for all kinds of discovery and improved knowledge, but we cannot use the priesthood (make and keep the covenants) or assume we can access His grace for perfection and sanctification (Moroni 10:32-33) on our own.
32 Yea, come unto Christ, and be perfected in him, and deny yourselves of all ungodliness; and if ye shall deny yourselves of all ungodliness, and love God with all your might, mind and strength, then is his grace sufficient for you, that by his grace ye may be perfect in Christ; and if by the grace of God ye are perfect in Christ, ye can in nowise deny the power of God [priesthood, sanctification by the Holy Spirit of promise].
33 And again, if ye by the grace of God are perfect in Christ, and deny not his power, then are ye sanctified in Christ by the grace of God, through the shedding of the blood of Christ, which is in the covenant of the Father unto the remission of your sins [perfection], that ye become holy, without spot [sanctification].