Do you have any Biblical references of this method of discernment of supernatural claims?
God gives you His Holy Spirit when you're genuinely saved.
John 14:16 And I will ask the Father, and he will give you another advocate to help you and be with you forever— 17 the Spirit of truth. The world cannot accept him, because it neither sees him nor knows him. But you know him, for he lives with you and will be in you.
With that said, the Holy Spirit helps you to test the spirits.
1 John 4:1 Beloved, do not believe every spirit, but test the spirits to see whether they are from God, for many false prophets have gone out into the world.
When Paul came to Berea, the Jews searched the Word of God to see if his claims were true.
God gave you His Holy Spirit, not just His Word. I think this is why you lack faith that these gifts of God are still in operation today. Have faith in the Holy Spirit who is also God to give you discernment at least, if you don't want to receive these gifts to serve God. There are so many other ways to serve Him!
But like the Bible says though, God doesn't give these gifts to every believer. It's okay if you don't have them, but you're not to prevent or discourage the people that God wants to give these gifts to.
1 Corinthians 14:39 Therefore, my brothers and sisters, be eager to prophesy, and do not forbid speaking in tongues.
If someone claims the ability to heal the sick I will not pray to know if he is truthful.
I will look for the holes in his roof. Luke 5:17-39
Yes, by all means. Test the spirits!