[sitecore-jss-nextjs] Rework promises in context implementation

[art-alexeyenko]

• This PR follows the Contribution Guide
• Changelog updated

Description / Motivation

Follow up to context changes.

• Ensure SDK init methods can throw without halting context init
• Ensure getSDK method can properly resolve and reject
• Ensure relevant message is passed when getSDK rejects

Testing Details

• Unit Test Adjusted
• Manual Test/Other (Please elaborate)

Types of changes

• Bug fix (non-breaking change which fixes an issue)
• New feature (non-breaking change which adds functionality)
• Breaking change (fix or feature that would cause existing functionality to change)

[illiakovalenko]

From my understanding this implementation doesn't really change an error handling behavior, or you didn't update unit tests accordingly. We need to discuss it further, maybe I missed some case
(Update: had a sync call and discussed concerns that appeared and changes to be made)

[illiakovalenko]

Very minor suggestion :)