certain extensions breaking due to WebSQL being disabled

[kjozwiak]

Description

It appears that some extensions are failing due to us disabling WebSQL as per brave/brave-core#4463.

Steps to Reproduce

1. install 1.6.56 CR: 80.0.3987.122 (Dev) or 1.7.39 CR: 80.0.3987.132 (Nightly)
2. install https://chrome.google.com/webstore/detail/speed-dial-fvd-new-tab-pa/llaficoajjainaijghjlofdfmbjpebpa/related?hl=en

Actual result:

Expected result:

Reproduces how often:

100% reproducible using the above STR.

Brave version (brave://version info)

Brave | 1.6.56 Chromium: 80.0.3987.122 (Official Build) dev (64-bit)
-- | --
Revision | cf72c4c4f7db75bc3da689cd76513962d31c7b52-refs/branch-heads/3987@{#943}
OS | macOS Version 10.15.3 (Build 19D76)

Brave | 1.7.39 Chromium: 80.0.3987.132 (Official Build) nightly (64-bit)
-- | --
Revision | fcea73228632975e052eb90fcf6cd1752d3b42b4-refs/branch-heads/3987@{#974}
OS | macOS Version 10.15.3 (Build 19D76)

Version/Channel Information:

• Can you reproduce this issue with the current release? No
• Can you reproduce this issue with the beta channel? No
• Can you reproduce this issue with the dev channel? Yes
• Can you reproduce this issue with the nightly channel? Yes

Other Additional Information:

• Does the issue resolve itself when disabling Brave Shields? N/A
• Does the issue resolve itself when disabling Brave Rewards? N/A
• Is the issue reproducible on the latest version of Chrome? N/A

Miscellaneous Information:

CCing @bsclifton @rebron @brave/legacy_qa

[kjozwiak]

Verification PASSED on macOS 10.15.3 x64 Catalina using the following build:

Brave | 1.7.71 Chromium: 80.0.3987.149 (Official Build) dev (64-bit)
-- | --
Revision | 5f4eb224680e5d7dca88504586e9fd951840cac6-refs/branch-heads/3987_137@{#16}
OS | macOS Version 10.15.3 (Build 19D76)

• Verified the original STR via certain extensions breaking due to WebSQL being disabled #8542 (comment) and ensured that the Speed Dial [FVD] extension worked without issues

Verification passed on

Brave	1.7.71 Chromium: 80.0.3987.149 (Official Build) dev (64-bit)
Revision	5f4eb224680e5d7dca88504586e9fd951840cac6-refs/branch-heads/3987_137@{#16}
OS	Ubuntu 18.04 LTS

Verified test plan from the description

Verification passed on

Brave	1.7.73 Chromium: 80.0.3987.149 (Official Build) dev (64-bit)
Revision	5f4eb224680e5d7dca88504586e9fd951840cac6-refs/branch-heads/3987_137@{#16}
OS	Windows 10 OS Version 1803 (Build 17134.1006)

• Verified the original STR via certain extensions breaking due to WebSQL being disabled #8542 (comment) and ensured that the Speed Dial [FVD] extension worked without issues