(i wrote this before seeing your bracket, i'll comment on that at the end)
i don't really like quad elim because early wins matter so much more than late wins. for example, a team with a bye only needs to win 2 games before they have 4 chances to qualify. new bracket for reference (here they are seeded 1 (a) to 12 (l) and the higher seed wins each time):
-----------------------------------------------------
el . de . . . . ad . . . . . . . . . . . ab . a
fk . cf . . . . bc
gj . bg
hi . ah
-----------------------------------------------------
. . . hl . eh . ce . . . . . . . cd . bc . b
. . . gk . fg . df
. . . fj
. . . ei
-----------------------------------------------------
. . . il . gi . eg . . . . . ef . de . cd . c
. . . jk . hj . fh
-----------------------------------------------------
. . . . . . ik . hi . gh . fg . ef . de . d
. . . . . . jl . gj
a team with a bye could go WWLLLW and qualify (3-3), while a non-bye team could go WLLWLWWWWWL (7-4) and not qualify.
naturally, the fix would be something swiss-like, like in the traditional 16-team swiss where each win and loss is equally important. however, this is of course difficult with only 12 teams, more than 3-5 games per team, and only picking 4 to advance. (for example, the T2 swiss stages, which require 4 wins to advance, have a weird 3-2 vs 2-3 matchup in the middle, and there would only be more of this when you deviate further.)
okay so now that i saw your bracket: i can't really understand what's going on here? two teams disappear in bracket 2, round 4; the arrows from bracket 2, round 5 point to nothing; and one team magically appeared in bracket 4, round 6. another issue arises with trying to fit our current 12 teams, since deleting 4 teams from this bracket has some bad effects down the line (e.g. two teams in bracket 3, round 4 get a bye while the others don't). however, i like how this addresses my issue of upper bracket matches "skipping to the right" a ton, as it did in my bracket, so if we can work this out that i am all for it!