@tholah1 You've asked a lot of very excellent questions, but they require a bit of a long answer, so bear with me.
There are two common issues when it comes to dips. Anterior shoulder pain, and costochondral pain. The latter is why they've not been included as standard.
Anterior shoulder pain is what I am referring to in the preamble of my tutorial on dips, and a majority of instances can be resolved with fixing one's form, with a minority being due to deeper issues that require some individual corrective exercise to rectify.
On the other hand, costochondral pain/problems exist on the flip side of that, where most examples do not seem to be form related and that require targeted corrective exercise to mitigate, in my experience both personally and observationally.
This means that some people can do dips with good form, avoiding anterior shoulder pain, but still get chostochondral pain. My current understanding of the most common cause aside from the same form faults that cause anterior shoulder pain (as they CAN co-occur due to those, like shoulder caving), is that it is something to to with a combo thoracic mobility and the muscles that control scapular tilting, which is kind of a tough and complicated area to work on and many people will not want to commit the time and energy, nor may they have the knowledge to work on this, especially when other exercises exist to get the job done perfectly well. I've observed this happen with a high enough frequency that it's called into question not my esteem for dips as an effective exercise, but their suitability as a standard blanket recommendation for a program designed for mass consumption.
For individuals that do not experience any sternum/costochondral issues from dips, or those who do but are willing to put the work in to fix the underlying issues causing the costochondral pain, e.g because they might wish to work on weighted dips in the future, the option to swap pushups out for them after the appropriate level of strength is there.
Hopefully that addresses your question about the relationship with dips and pain, with the tl;dr of:
'Dip pain is not common solely because of bad form'
So that's reason 1 of why they exist less prominantly in the program the way they currently do.
Reason 2 is that 3 push exercises would be too many for a full body beginner training program. This means you can either have:
- Dips and Push-ups (Two exercises that work basically the same muscles in slightly different planes, akin to bench press and decline bench press.)
- Push-ups and Pike/HSPU (Two exercises that serve functionally distinct purposes, akin to bench press and overhead press)
- Pike/HSPU and Dips (Two exercises that serve functionally distinct purposes, like decline bench and overhead press)
Option A is not an efficint use of two exercises and is missing an overhead pressing component which is an important component of a training program
Option B is the best standard for everyone starting the routine because push-ups have a lower strength requirement
Option C is good for people who have built up their pushing strength a bit and want to focus in on dips, and do not experience issues with dips.
Once an individual graduates to a more advanced program like a dedicated push/pull (rather than the push/pull modification to this routine), one could reasonably do all 3 together on a push day. But that's not appropriate for this programs purpose/demographic.
Hopefully that makes sense. I will address the birddog question in a separate comment to avoid this being too long.