Running a rod thru the flues to clean them might damage the flues.
But you won't know that until later on......
------------------------
A proper repair would be to take the top and bottom off the
radiator, and pressure-flush out the clogs without damaging the flues.
.......and once it's back together, you won't know if it was done properly until further down the road.

A second choice would be to remove the clogged flues and install a new, unclogged flue....or less labor to replace the whole radiator core.
........and I expect them to give back the replaced one.......I want to look at it.

Either solution is labor intensive and the rad shop needs experience
working with aluminum...........I suspect most shops want to get a rad in
and send it out as fast as possible.....and some shops I have been to
1.don't fix it the way I ask them to and/or 2.don't fix it the way they say they will.....haven't had good experiences......

There might be another aftermarket aluminum radiator company
that sells for less than Mazda Comp.........not recalling it right now
....used to be in St Pete...........maybe do an internet search.......
Let us know what you decided, and how it worked out....

What does a Mazda Comp radiator sell for?-Bob