I think I've finally figured out why I've been finding some aspects of your argument so dissonant - it's that you're conflating the individual privilege of many (not all) life peers that led them to being made life peers, and the institutional privilege enjoyed by the Church of England in the Lords.
Life peers certainly come from disproportionately privileged backgrounds (57% were privately educated, etc). There is certainly a lack of a transparent and fair selection process to ensure that the Lords reflects the society it serves. I think it's a dreadful institution that should be scrapped.
But for me, claiming that the individual privilege of many/most peers is the same as having institutional privilege for representatives of one certain group is where the no more/less privileged argument falls down, given that privilege seems to be the absolute sticking point for you.
Because unlike the life peers, where depending on your politics, profile and/or achievements people from all backgrounds are technically eligible to be life peers, there are two groups of peers that have institutional privilege rather than individual: the life peers (qualifier: was your dad a lord and all the other kids of lords said you could be one) and the bishops (have you risen up the ranks of the right church, ok you can have a say in our laws.)
The question of institutional versus individual privilege is the question for me here because there's an immediate power imbalance. Sandi Toksvig has a profile that enables her to get attention; she comes from a relatively affluent background. So sure, there's a degree of individual privilege. Regardless of Welby's individual privilege he enjoys a degree of institutional privilege - and therefore power - that Toksvig does not have.
She's not punching down. She's not even one person with privilege and a platform challenging another person with privilege and a platform. She's a person challenging the privilege and platform of an institution that enjoys a degree of privilege in our constitution denied to all other groups except a handful of hereditary peers.