It sounds to me like she is exercising the limited control she has. Granted, other things may be going on here. The ability to interrupt play, for instance, is a different skill from the ability to identify that you need to go to the bathroom. It is possible to have acquired one without the other, but you'd need both to avoid accidents, in certain scenarios.
Even so, giving time outs for wetting herself sounds to me like an extremely authoritative approach, and potty training is one of quite few areas in a young child's life where the child is ultimately in control. If cooperation breaks down, the parent cannot just override the child and demand to have their way. I'm not surprised if she's simply not willing to yield to her parents' will in this.
So I would suggest stop the shaming and the punishments. Stop the nagging (it's not working, as you've noticed), and stop the rewards (she needs to desire to be potty trained, not desire a reward). Stop all of that, and even if you don't succeed with anything else, don't pick it up again, because none of it's working and it's annoying and damaging for all involved. You need instead to appeal to her own motivation for the desired behavior.
Ask her, in a non-judgmental manner, how she feels when she's wet herself. The negative consequences she associated with wetting herself shouldn't be anything external - certainly not punishment, but also not shame or belittlement. Because the logical thing to a child would then be to focus on seeking other outcomes. Remove all of that so that the remaining negative consequences are purely natural: the inconvenience, the unpleasant feeling, the red butt, as you mention. She needs to associate wetting herself with consequences that can only be addressed by a change in behavior, and I believe these are now being overshadowed by harsher consequences such as time outs.
Once she has internalized the desire to potty train, and is not just intent (or not intent) on complying with your desires, ask her what kind of help she would need from you. If she requests reminders or rewards, those may again be on the table. She probably knows why she says she doesn't need to go when you ask her, even if she isn't able to verbalize her reasons, but if you ask her when or how she wants to be reminded, she may be able to answer. She may be fully aware that she won't be able to interrupt an exciting activity to go to the bathroom, for instance, but desire help in being reminded to routinely go before or after key activities.
You may be able to get her on board with routine bathroom visits, say every morning, after lunch and before you leave the house. If you explain how preemptive bathroom visits can avoid having to interrupt play, she may accept that these are beneficial, and you could reach an agreement that in these scenarios, she should be reminded, and she should go even if she doesn't feel a need to.
The solution may also be something entirely different. Again, she may know why without being able to verbalize it, but may still be able to come up with solutions on her own that will be better than yours. Just taking a step back and showing that you're not going to force your way on this may be enough, as she'll realize it's also her own desire which she may not have had a chance to consider.