Explanation:

The correct way to avoid full table scans here is to add a computed column that extracts the JSON scalar property with JSON_VALUE , and then create a nonclustered index on that computed column with the report's returned columns in the INCLUDE list. Microsoft's JSON indexing guidance specifically recommends creating a computed column that exposes the JSON property you filter on, using the same expression as in the query, and then indexing that computed column.
So the computed column must be:
AS JSON_VALUE([log], ' $.severity ' ) PERSISTED
This is correct because $.severity is a scalar JSON value, so JSON_VALUE is the proper function.
JSON_QUERY would be for extracting an object or array, not a scalar property. Microsoft also notes that persisted computed columns can improve access speed for JSON-derived values.
The index should then include:
INCLUDE (LogId, LogDateTime, [log])
That is the right covering strategy because the report filters by severity but returns LogId, LogDateTime, and log. Microsoft's guidance on included columns explains that nonkey included columns let a nonclustered index cover more queries and reduce extra lookups to the base table.
So the completed code is:
ALTER TABLE WebSite.Logs
ADD severity AS JSON_VALUE([log], ' $.severity ' ) PERSISTED;
GO
CREATE INDEX ix_severity
ON WebSite.Logs(severity)
INCLUDE (LogId, LogDateTime, [log]);
GO