@how gai plus zatR makes gAyantam when zatR links to zakunim

gai गै together with /zatR makes a nounbase **gAyat- गायत्.

gai गै + /zatR
→ gai गै + /zap + /zatR
→ **gAya- गायॱ + /zatR
→ gAyatR- गायतृॱ by atoguNe
→ !**gAyat- गायत् "that is / are singing"

This **gAyat- गायत् has /Rdit, inherited from /zatR.

Now when we make this nounbase link to **zakunim शकुनिम्, which is a masculine noun and has /am at the end, we get, it must copy the gender and /sup of whatever it linksto, so we get —

**gAyat- गायत् m + /am
→ gAyant गायन्त् + /am by ugidacAMs..., because **gAyat- गायत् is /Rdit
→ !**gAyantam गायन्तम् "him whom is singing"

What if, instead of linking to **zakunim शकुनिम्, it were linking to **mayUrIm मयूरीम्?

**gAyat- गायत् f + /am
→ **gAyat- गायत् + /GI + /am by jAterastrIv...
→ gAyant- गायन्त् + /GI + /am by zapzyanornity...
→ !**gAyantIm गायन्तीम् "her whom is singing"

Why does the sUtra say aprathamA अप्रथमा "linking to a non-first"?

This sUtra allows us to glue **gAyat- गायत् to the **zakunim शकुनिम् word in zakunim pazyAmi शकुनिं पश्यामि "I see bird" because **zakunim शकुनिम् carries a non-first ending. But not to the first-ender **zakunis शकुनिस् of zakuniH patati शकुनिः पतति "bird flies" and zvetaz zakuniH श्वेतः शकुनिः "bird is white".

Why did we use /zatR in the examples and not /zAnac?

Because /tip is flat. /zatR is flat and /zAnac is bent.