When I killed tray after tray of seeds, I blamed overwatering and soiless mix that held too much water and not enough air, especially on the surface. Maybe "peat" needs to have much loger fibers, or LOTS or Perlite and grit to "open it up". Soiless mixes that I buy seem to be mostly powdered peat, which has the consistency and aeration of pudding when damp.
Then I started adding lots of shredded, screened pine bark (mulch) and some grit to my seed starting mix. Excess water drains away quckly, the surface dries out much faster, and I have NO damping off any more. My theory is that the roots need air, and if the soiless mix has no gaps for air space, the roots drown.
(Or I could learn how to avoid overwatering. I haven't mastered THAT after 3-4 years of trying. But laying down some cotton flannel on the bottom of the tray (for pots & cells to sit on) helps me to overwater LESS. It wicks excess water out the bottom of cells, and carries some water to the bottom of cells that started to dry out first.)
However, small seeds like petunias seemed to fall down into the gaps between the bark, and fail to sprout. This year, I'll use a finer top layer (like vermiculite) for the very smallest seeds.