It's still LCOE, which is by definition focused on generating assets. It doesn't track societal costs or costs offloaded to other entities, like grids or households, that can be attributed to a particular generating technology becoming dominant.

Things like sky high electricity prices during that blackout in Texas a few years ago don't really reflect in LCOE.

That particular firming calculation on page 8 (if that's the one you have in mind) is a bit misleading, they use average capacity instead of historical capacity factors [1], which doesn't make any sense in places with seasonal variability. You need waaay less storage to "firm" an energy source that is active 25% of every day vs one active only during the summer. Time distribution matters, and they do acknowledge it in small print in a footnote, suggesting that they only mean firming the "duck's head" in the duck curve [2]. Which only captures a tiny part of a system that would need to store a whole night (or season's) worth of energy.

[1]: "ELCC is an indicator of the reliability contribution of different resources to the electricity grid. The ELCC of a generation resource is based on its contribution to meeting peak electricity demand. For example, a 1 MW wind resource with a 15% ELCC provides 0.15 MW of capacity contribution and would need to be supplemented with 0.85 MW of additional firm capacity in order to represent the addition of 1 MW of firm system capacity."

[2] "For PV + Storage cases, the effective ELCC value is represented. CAISO and PJM assess ELCC values separately for the PV and storage components of a system. Storage ELCC value is provided only for the capacity that can be charged directly by the accompanying resource up to the energy required for a 4 hour discharge during peak load. Any capacity available in excess of the 4 hour maximum discharge is attributed to the system at the solar ELCC."