GrouchyGrape my recommendation (although many straight refuse this) is to carry a second phone purely for work
I wouldn't dismiss this out of hand, but it does seem pretty extreme. It's something I might have to consider if there is no urgency to fix these issues, or I can't find a better app.
I would think these basic phone functions would not only be appreciated by anyone using the phone app, but actually expected. Call merging and answering incoming calls while on a call should be a priority to have working properly. The rest of the issues I brought up are frustrating and inconvenient, but don't break the core functionality of the phone.
Thanks for the suggestion.