can i make the update part of this query use a "with recursive" expression?

currently-working.sql

-- for testing, completely reset highest_child_id columns
update calls set highest_child_id = NULL;

-- find rows where no other row has the row's id as its parent_id
-- those shall seed the progress with initial "highest child id"
-- values
with no_children as (select id from
	(select c.id as id, count(children.id) as childcount
		from calls c
			left join calls children on c.id == children.parent_id
		group by c.id)
	where childcount == 0)
	update calls set highest_child_id = id where calls.id in no_children;

-- set a row's highest_child_id to
update calls set highest_child_id = (
    -- the maximum of its children's highest_child_ids
		select max(children.highest_child_id) as maxval
			from calls children
			where children.parent_id == calls.id
			group by children.parent_id
		)
  -- unless it is already set. also, it must
	where highest_child_id is null and calls.id in (
		select foo.id from
      -- correspond to a third set of rows
			(select parents.id as id, count(children.highest_child_id) as nonnullcount, count(children.id) as allcount
				from calls parents
					left outer join calls children on parents.id == children.parent_id
				group by parents.id) foo
    -- where every child has its highest_child_id already set.
		where foo.nonnullcount == foo.allcount);

-- display the results so far
select c.parent_id, c.id, c.highest_child_id, group_concat(child.id, " ")
 from calls c left outer join calls child on c.id == child.parent_id
 group by c.id
 order by c.id;

schema.sql

CREATE TABLE calls(
  id INTEGER PRIMARY KEY ASC,
  parent_id INT,
  
  -- can ignore everything after this point, especially the foreign key.
  routine_id INT,
  osr INT,
  spesh_entries INT,
  jit_entries INT,
  inlined_entries INT,
  inclusive_time INT,
  exclusive_time INT,
  entries INT,
  deopt_one INT,
  deopt_all INT,
  rec_depth INT,
  first_entry_time INT,
  highest_child_id int,

  FOREIGN KEY(routine_id) REFERENCES routines(id))

sql-stuff.sql

update calls set highest_child_id = NULL;

with no_children as (select id from
	(select c.id as id, count(children.id) as childcount
		from calls c
			left join calls children on c.id == children.parent_id

		group by c.id)
	where childcount == 0)

update calls set highest_child_id = id where calls.id in no_children;

-- repeat this a whole bunch, or find a way to make it recursive
-- of course this currently only selects the rows that can currently have their highest_child_id set,
-- it would need to be an update statement for this to work.
select c.id as id, max(children.highest_child_id) as max_child_id
	from calls c inner join calls children on c.id == children.parent_id
	group by c.id;