I dont think getting the gun up to see where its pointed is a nonstarter at all. I have to say Ive always been baffled by the idea of shooting stockless shotguns solely from the hip rather than getting them up where you can see where its pointed, unless one is talking contact type distances and concerned about it being taking away from you. The common vertical style shotgun pistol grips never appealed to me after I shot a few however.
I dont think its accurate to say the birdshead grip meant to increase length, I think it was used in the case of the shockwave for that reason, they used an existing commercially available grip to achieve that. Theyve been around a long time in the form of shortened buttstocks, as well I believe as custom made examples. The cut down single barrel I had way back when being one common type of example, and it was not difficult to shoot held out to see the barrel and hit cans in the air (all I had for airborne targets at the time).
Edit: I recall the Witness Protection guns from the 1980s as well as home and old gunsmith chop jobs from as far back as the late 1800s with similar grips. I recall articles on the witness protection guns from way back, but few details. One I just searched (2017) had a picture showing holding the gun up to eye level to shoot, mentioning that shooting from the hip was a great way to totally miss the target even at relatively close range.
https://www.tactical-life.com/firear...rotection-870/