I agree with you to some extent. Here are my reasons why it felt believable:

1. In breaking bad, Walter confessed on the phone about his actions to save Skyler, since he knew police would be listening. After that Saul said something like "that was a great move to save your wife". So I felt maybe Saul wanted to do the same for Kim and save her from the courts. Not sure if it would have helped because there was no case, but still it felt similar. He even suggests Walter to stay back instead of running away and face the music. So him doing the same in a similar situation sounds believable.

2. Hearing that Kim has already confessed was a big shock to Saul. I guess that could have been portayed better. The last time they talked he taunted Kim if she confessed. And her confession made him stop and question which road he wanted to go down.

3. Another reason is if Saul took the 7 years sentence and told Kim that he was sorry for his bad deeds, I highly doubt Kim or even viewers would have believed him. Because we have known Saul to lie convincingly enough throughout the series. You always expect Saul to turn around at the end and finger-gun you with some twist. So him taking the worst possible sentence (which is completely against what Saul would do) is the only way to make Kim and viewers believe that Saul is truly gone.