Confused regarding variational principle derivation

I am reading the following article, https://arxiv.org/abs/1812.08767, and I am trying to go though the derivation of McLachlan’s variational principle for imaginary time evolution. The derivation is done in section B.1.2.

I am confused about going from this equation:

to this equation (assuming real $theta_i$):

I think the $x$ at the end is a typo. What I dont understand is how the variation $delta$ removes the need to sum over $i$, going from eq. (120) to (123). They also remove the $dot{theta}_i$ factor and seem to replace it with $delta theta_i$. I also dont understand why the final term in eq. (120) disappears.

I think I am overall a little confused as to what the variation $delta$ actually does to an equation. Could anyone explain?