If Joffrey or all of the kids were Robert's, the entire dynamic changes! Long answer as there is a lot to unpack.
First of all, Cersei would have to be a different person to allow herself to have Robert's children. Would this Cersei love her children anyway or would her resentment trickle down to them as they would be a living embodiment of her lack of agency? I know she spits out some bullshit to Sansa about how she might not love her husband but she will love his children, but Cersei most certainly made sure she will never birth any of Robert's children, even going as far as aborting one of them. So Cersei herself would have to be somewhat different to begin with for the children to be Robert's.
But lets say Cersei is a different person and she allows herself to birth Robert's children and she still loves them. It's very likely Joffrey ends up the same way. Please understand that Joffrey is very much a product of both of his parents and Robert would not suddenly have a personality transplant just because his child is a brunette. He was decent to Mya because he was a pre-war teenager at the time and on good terms with her mum. He is not on good terms with Cersei and as much as he is an absentee dad, Cersei also plays a big role in keeping her children away from him. As long as Robert has children with Cersei and both him and Cersei are not completely different people from how we know them then Robert will not be involved in the children's lives. Be it out of spite or because Cersei sees the children as extensions of herself still, she will keep him away, and he in return will not take much interest in them.
If Cersei resents her children then Joffrey is not raised primarily by his mother and Robert might be more involved purely because Cersei is not blocking him or might still not care much. Either way, this Joffrey has more chance of being more like Tommen and Myrcella, who by all accounts we have seen to be decent people. So the only scenario where Joffrey has a chance of being a decent person is one where Cersei is not involved in his upbringing.
However, at this point the political landscape looks different. Cersei has ensured the stability of this Lannister-Baratheon dynasty. Jon Arryn has no reason to suspect foul play and likely does not die. Yes, Lysa would still have a reason to get rid of him, but would Littlefinger? Littlefinger is the one who devises the plan, Lysa executes it. He needs Jon Arryn dead in the main series as he is not ready to have the Lannisters outed, so if Jon Arryn is not about to topple the Lannisters then does Littlefinger have a reason to get rid of him? Without Jon Arryn being dead the position of hand is not available. Ned Stark is not in KL. Cersei has no reason to get rid of Robert when she did. Renly's plan of supplanting Cersei with Margaery also does not make much sense when there are 3 trueblood children around and there is no reason to suspect them not to be Robert's. Stannis would not think himself the rightful heir. Renly would not think the throne is ripe for the taking. The North does not declare independence. The Others/Daenerys/Aegon find Westeros in a somewhat better shape. Varys would still try to destabilise things, but would need another strategy. Littlefinger would also still be there ready to take advantage of the chaos.
Lets move this one step further then. Lets say Jon Arryn does die and the Starks and the others suspect foul play was involved and that the Lannisters had some other reason to want him dead. Lets say Cersei kills Robert for other reasons as well, such as wanting to be free of him now that she has a child who is close to his majority. Maybe she wants the regency and to enjoy her time in the sun. Stannis still would not think of himself as the rightful heir. Ned would not think of betraying Robert's son but would be weary of the Lannisters. The most likely scenario is that Tywin shows up and removes Cersei from the picture and takes main control of Joffrey's upbringing just like he meant to do in the main series. Ned does not back down because he thinks he has a duty to protect Robert's heir and keep the Lannisters in check. The Lannister have no excuse to declare the Starks as traitors. Potentially Tywin agrees to a shared regency a la Aegon III. Ned would still call for Stannis, but this time for support due to his suspicions that the Lannisters had Jon Arryn and Robert killed. Stannis would probably be part of the shared regency.
So now you still have a Lannister-Stark power struggle but it's over Robert's heir. There is no War of the Five Kings. The Tyrells will be pushing hard for a Margery-Joffrey match knowing of the skirmishes between the Lannisters and the Starks and how Tywin might not want to have a Stark queen. If Stannis is convinced of some Lannister foul play as well he allies himself with the Starks in the attempt to diminish the Lannisters' power in court and over the heir.
So you could end up with Lannister-Tyrell vs Stark-Baratheon skirmishes going on until either Daenerys or the Others come knocking on Westeros' door.