Do it like this

INSERT INTO people
(run, first_name, last_name, dob, address, insert_date, processed_flag)
SELECT 1, first_name, last_name, dob, address ,'2023-07-19', 2
FROM #peopletmp

There are several ways to split it into 8 different processed flags depending on your rules. Here are two examples you could adapt

-- generate some demo data
declare @demo table (dname varchar(255), dtype varchar(255));
insert into @demo (dname, dtype) select [name],[type] from sys.objects;

-- divide into 8 groups 1, 2, 3, 4, 5, 6, 7, 0 based on the order of one of the data items
select dname, (ROW_NUMBER() over (order by dname)) % 8 as groupNumber
from @demo;

alternatively, and probably the more likely scenario...

-- generate some more demo data 
declare @types table (groupNumber int identity(1,1), dtype varchar(255));
insert into @types (dtype) select distinct [type] from sys.objects;

-- divide into 8 groups 1 through 7 based on some arbitrary rules
select d.dname, d.dtype, t.groupNumber
from @demo d
inner join @types t on d.dtype = t.dtype;

Finally, another technique based on first letter of a name

-- Generate some more demo data based on alphabet
declare @alphas table (alpha char(1), groupNumber int);
insert into @alphas (alpha)
select top 26 CHAR((ROW_NUMBER() over (order by name)) + 64) 
from sys.objects;
-- Break the alphabet into 8 arbitrary groups
update @alphas 
set groupNumber = CASE 
        WHEN alpha IN ('A','B','C') THEN 1
        WHEN alpha IN ('D','E','F') THEN 2
        WHEN alpha IN ('G','H','I','J') THEN 3
        WHEN alpha IN ('K','L','M') THEN 4
        WHEN alpha IN ('N','O','P') THEN 5
        WHEN alpha IN ('Q','R','S') THEN 6
        WHEN alpha IN ('T','U','V','W') THEN 7
        WHEN alpha IN ('X','Y','Z') THEN 8
    END;

-- Group into 8 categories based on 1st letter of name
select d.dname, d.dtype, a.groupNumber
from @demo d
join @alphas a on a.alpha = LEFT(d.dname,1);