I agree, the nature of the convictions should be a factor. I also agree that a sort of "cooldown" from a conviction would be reasonable, before having eligibility for holding political office restored. I've been leary of the simplified "convicts shouldn't hold office" statement though, since the original intention of that lack of disqualification criteria was, to my understanding, to prevent political imprisonment from barring opponents from holding office. That seems like the sort of thing the current administration would jump on if they could, as well.
That was due in large part to the fully suited scenes being cut from the orginal show. The characters we know in the English version are really a separate cast.