Honestly, when I give pills to chickens I just get it in their beak and they swallow. I don’t even try to go to the side as I don’t put it very far back. The tube feeding… Absolutely, yes I go to the side.You should read up a bit and look at some pictures until you feel confident - when I get a minute later I will post some for you - but there are lots on this site.
Basically the airway is in the middle at the back of the tongue and the oesophagus is behind and goes on both sides and is more oriented to the chicken's right.
So, if you put something to the side on the chicken's right it will go down into the crop.
Once you have popped the thing in let the chicken move its head freely and it will 'adjust' things.
@micstrachan is really the expert here so she can probably describe it better. I like to think of it as putting it inside the chicken's cheek - which is nonsense because they don't really have cheeks but that is what it feels like to me!
The key is to keep it to the side and then let her handle it from there!
Edited to add: Of course the real key is having a treat immediately handy to say thank you for being a good girl!
However, you described it quite well.