What stands in the way of Christians today from speaking in the name of Jesus and healing people as Peter and John did? Is our faith too small or are medical advancements proof of His miracles?
I wouldn't say medical advancements are proof of God's miracles. Remember, Luke, was a medical doctor. So he's telling us that miracles are different than medical advancements (though we're thankful for those advancements!). But there are a few reasons why we may not be seeing healings like we do in the book of Acts.
First, there is definitely something distinct about the miracles we see done by the apostles (Peter and John and the others). Acts 2:43 connects the miracles to the apostles (not the average Christian). And these miracles were proofs of their message of the resurrection of Christ. Second, though we may not expect to see the miraculous today, in the same degree that we see happening through the apostles, I think one reason why we don't see healing often in the church today is because we both misunderstand what the gift of healing is and don't pray for others to be healed (or asked to be prayed for) with very much regularity. I've had the opportunity to pray for a few individuals at Gateway who were blessed by God to be healed. I'm thankful for those moments as it reminds me to continue praying for others to be healed and to pray in faith, believing that God does heal today.
One final note about healing. Though I believe God does heal today, I also believe that He uses suffering for our sanctification (to make us holy). It's easy to desire comfort over holiness. But suffering (or illness) can be used for our good and growth in holiness. For this, we should praise God.