The F-15E was developed privately before the USAF issued the ETF tender. The MQ-9 was developed privately by General atomics before they anticipated the USAF's desire for an improved MQ-1.PLAAF is the boss,I don't buy the rumor that the SAC tried to collect the R&D money by its own connection in the AVIC since if the boss won't buy its product,it's useless to spend money and energy on such a project.And on the contrary,I believe that the SAC J-XX project has a high priority in the PLAAF's future plan if it's not more important than the J-20 due to the reasons:
1.The PLAAF intended to buid a both defensive and offensive force(by recent PLAAF publictions),the SAC J-XX fighter/bomber will be the backbone to project the air to ground firepower in an offensive mission.
2.If the SAC J-XX is cheaper enough(at least it seems so from current leaked information) than the J-20,mostlikely the PLAAF will buy more of it than the J-20.That means the PLAAF will form a high/low combination by the J-20/SAC J-XX.
3.I can smell the distrust from the PLAAF to the CAC.The SAC J-XX has a role to be the plan B that if the J-20 failed,the PLAAF still has another project to ensure the sucess.
Overall,the SAC J-XX services the PLAAF's future offensive plan and it's important to the PLAAF,the SAC project could not to be a self-funded one.It's important,if not more important than the J-20.
Privately developing a new aircraft in the early stages and then having the military become interested is not new at all.
I agree with your points 1 and 2. SAC's J-XX could certainly be used in those roles but that does not support the idea their "J-19" is being privately or government funded.
Point 3 is ridiculous -- if there's any distrust it's PLAAF towards SAC. The fact that they've only recently picked up J-11B/S and J-15 production is probably because PLAAF were commited to the platform as the current mainstay of their heavy fighter force. The same way they are commited to the J-20 as their future heavy fighter force.
SAC's J-XX proposal lost for a reason, and CAC has shown themselves more than capable to take on the task of developing multiple new fighters in recent years.
SAC's J-19 is probably not being funded by the air force.