Both the steel rake and the 2x4 sound good to me. I use the flip side of my
steel rake for leveling out soil in my garden and amending parts of my lawn.
I also use it in spring to rake out the dead grass from winter. The
investment is worth it!

Because you are using the flat side of the landscape rake to spread
amendment, you are using it just like using a 2x4. Good. I was under
the wrong impression that I should be using the teeth side of the
landscape rake. Now, I understand that I should be using the flat side
of the landscape rake. Thanks.

The 2x4 should be 4 to 6 feet long with rope tied to both ends. Any longer
than 6 ft. strikes some vague memory of something I heard about the
unweildiness of more length. Maybe it's a physics thing..I dunno.

The 2x4 idea should work. But I will have to calibrate the weight of
the 2x4 and mow the grass to the right height; otherwise, I will not
be able to spread the amendment to the correct thickness. Seem like
using a landscape rake is easier in term of adjusting the thickness of
the amendment.

The tire seems like too much weight and too little area treated.
Am I making sense?

I don't see how I can use a tire to spread amendment. But this is
irrelevant anyway because I don't have a spare tire.

Use the teeth side to rake the soil out after leveling it and to rake the
seed in.

I am not exactly sure how I can use the teeth side of the rake to rake
the soil out. Would you please describe the way that you do this?
Thanks.

Jay Chan