The point of the matter is not whether the power is real or not, strong enough to become noticeably undemocratic or not. The point is al those seats are not elected. Kudos to the House of Commons for trying to reform the upper chamber, but the fact remains you have a structure that is anchored in a deeply undemocratic past. Even if all the Lords Spiritual had a purely formal role (which they don't - they have the right to vote) this would still be unacceptable in a democracy. The members of the parliament are all supposed to be elected officials whose function should be to serve the society. They are paid officials and their seats cost money in ways that go well beyond their salaries. Having a guaranteed seat in the parliament is not only an issue of power, it's also an issue of guaranteed social status, monetary benefits and indirect political influence.
The very fact that you take such a lax attitude towards this reminiscent historical injustice speaks volumes, and I'm not just talking about you, this is an attitude prevalent in the UK (I travel for business a lot there, more than half my business is with the UK). I'm sure it's not a pressing need to rectify it because the size of the issue has been diminished over time. Here's hoping it goes away completely, but I believe you can't get rid of such an injustice completely as long as you cling to the image of a hereditary monarchy as being something good; monarchy is a huge injustice in and of itself, even if you somehow manage to isolate it from political power completely.