[Reland] WebUI: Enable @typescript-eslint/require-await check in build_webui().

Reland notes: This is the 2nd part of the reland of the reverted CL at
crrev.com/c/6253018. The CL was incorrectly reverted, since the problem
was limited in the local dev environment of the person that was having
the issue, where third_party/node/ for some reason was stuck in v16 (a
very old version) instead of v22.

Enabling `enable_type_aware_eslint_checks` by default in build_webui().

Bug: 394634491
Change-Id: I35f77e4d3aef7af08ad06f1881d079e184c8153d
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/6254799
Reviewed-by: Rebekah Potter <rbpotter@chromium.org>
Commit-Queue: Demetrios Papadopoulos <dpapad@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1418856}