Count Most Recent Streak

Is this view always within one calendar year, or does the solution need to take into account that a streak can continue from week 52 to week 1 of the next year?

And would it be an option to calculate the longest streak per week and person at data refresh time in Power Query? Or are there other attributes that you need to filter interactively?

Looks like your code is counting all week numbers (including partial weeks at the beginning and end of year, according to how WEEKNUM() works). This indicates a missing relationship between your date table (the one that provides the week number column titles) and your fact table (the one that provides the Billable values).
If this is the case, you need to create a virtual relationship in your DAX code.
This code works in my file, without a week column, after reproducing your observation:

Longest Streak (date based) = 
VAR _PersonsAndStreakLength =
    ADDCOLUMNS ( 
        SUMMARIZECOLUMNS (
            'Name'[NameKey]
        ),
        "@StreakLength",
        // get the weeks and their WeekOver80 flag
        CALCULATE (
            VAR _WeeksAndStreaks =
                CALCULATETABLE (
                    ADDCOLUMNS (
                        SUMMARIZECOLUMNS ( 
                            'Date'[Date]
                        ),
                        "@WekOver80",
                        VAR _Thursday = [Date]
                        VAR _StartDate = _Thursday - 3
                        VAR _EndDate = _Thursday + 3
                        VAR _WeekDates = DATESBETWEEN ( 'Date'[Date], _StartDate, _EndDate )
                        RETURN
                        CALCULATE ( 
                            COALESCE ( [WeekOver80], 0 ),
                            TREATAS ( _WeekDates, 'Streak'[Date] )
                        )
                    ),
                    WEEKDAY ( 'Date'[Date], 2 ) = 4
                )
            VAR _LastWeekWithoutStreak = MAXX ( FILTER ( _WeeksAndStreaks, [@WekOver80] = 0 ), [Date] )

            VAR _StreakLength = COUNTROWS ( FILTER ( _WeeksAndStreaks, [Date] > _LastWeekWithoutStreak ) )

            RETURN

            _StreakLength
        )
    )

RETURN

MAXX ( _PersonsAndStreakLength, [@StreakLength] )

The relevant change is this part:
CALCULATE (
    COALESCE ( [WeekOver80], 0 ),
    TREATAS ( _WeekDates, 'Streak'[Date] )
)