Second, who ever said that what occurs in Purgatory has nothing to do with Jesus Christ? Nobody.
It has EVERYTHING to do with Jesus Christ. The souls in Purgatory are in the state of sanctifying grace--i.e., they have the indwelling of the Holy Trinity, the life of charity, a supernatural, transformative relationship with Jesus Christ.
If the believer, who believes in Christ, the Christ who died for our sins and nailed them to the cross as a one time sacrifice for all of our sins, is not cleansed by his belief in Christ, as described in the NT, then nothing remains that can forgive sins.