Regeneration is only for the New Covenant, which the Ezekiel Passages are referring to - when God will make a New Covenant with His people.
The Old Covenant saints were saved by faith as well, but they did not receive the regeneration of the Spirit.
You cannot accept this because you were told that no one can believe until they are made spiritually alive first. But that is not what the Scriptures teach us.
The Spirit was always active in the world and enlightened those who believed in God, but the regeneration of God's indwelling Spirit did not occur because Lord Jesus had not yet paid the ransom for all sins.
Romans 10:17 (WEB)
17 So faith comes by hearing, and hearing by the word of God.
The Word of God is the Sword of the Spirit (Ephesians 6:17). The Word and Spirit working together enlighten (John 1:9), convicts (John 16:8-12), draws (John 12:32) the world (Ephesians 6:17; Hebrews 4:12).
Many resist the Spirit and the Word and remain condemned:
Acts 7:51 (WEB) 51 “You stiff-necked and uncircumcised in heart and ears, you always resist the Holy Spirit! As your fathers did, so you do.”
Isaiah 63:10 (WEB) 10 But they rebelled, and grieved his holy Spirit. Therefore he turned and became their enemy, and he himself fought against them.
Those who listen, the humble, will be convicted, enlightened, and drawn to Lord Jesus and the salvation He offers them through the Gospel (John 3:14-18), and so are the ones taught by God.
Psalm 25 (WEB) 9 He will guide the humble in justice. He will teach the humble his way.
Those who humble themselves before His Word and Spirit are the ones that God elects to save and to give to the Son. Those God foreknew, God also predestined for salvation, calling them through the Gospel - This promise is for all those who would love God (Romans 8:28).
Psalm 18 (WEB)
27 For you will save the humble, but the haughty eyes you will bring down.
By faith, in the New Covenant, we are Spiritually made alive when the Spirit indwells us by faith:
John 7:38-39 (WEB) 38 He who believes in me, as the Scripture has said, from within him will flow rivers of living water.” 39 But he said this about the Spirit, which those believing in him were to receive.
Romans 8:9-10 (WEB)
9 But you are not in the flesh but in the Spirit, if it is so that the Spirit of God dwells in you. But if any man doesn’t have the Spirit of Christ, he is not his. 10 If Christ is in you, the body is dead because of sin, but the spirit is alive because of righteousness.
By faith, we pass from spiritual death to life
John 5:24 (WEB) 24 “Most certainly I tell you, he who hears my word and believes him who sent me has eternal life, and doesn’t come into judgment, but has passed out of death into life.
By faith, demonstrated by following Him, we are turned from darkness to light.
John 8:12 (WEB) 12 Again, therefore, Jesus spoke to them, saying, “I am the light of the world. [Isaiah 60:1]. He who follows me will not walk in the darkness, but will have the light of life.
John 12:46 (WEB)
46 I have come as a light into the world, that whoever believes in me may not remain in the darkness
By faith, confessing our sins, we are washed, cleansed, from all unrighteousness.
1 John 1:9 (WEB) 9 If we confess our sins, he is faithful and righteous to forgive us the sins, and to cleanse us from all unrighteousness.
Acts 22:16 (WEB) 16 Now why do you wait? Arise, be baptized, and wash away your sins, calling on the name of the Lord.’
Blessings