You can be flogged standing in the middle of the room, laying on a table, or hanging from your ankles as you swing (moving target). But none of that has to be submissive.
Your question implies that the act of being tied to something is being submissive. But this is not true, being tied is being tied there is no way to understand the intent behind it without asking the person. There are many labels int he BDSM world. Sometimes to our detriment.
In the simplest terms the action in most play scenes can be described as having a top performing an action on a bottom. If there is no mental dynamic in the scene this is where it ends. However if there is a dominate/submissive component that can add to the scene, but the matchup is not necessarily top=dom & bottom=sub. T dominate in the relationship may like to get flogged and so they take the bottom position and have their submissive flog them. The dominate can even be tied to a cross, the act does not decide the mental state, it’s the mental state that decides the act.
I have a friend on Fet who lists “Having the, ‘I’m a bottom and not a submissive conversation.'” as one of her dislikes on her profile. She loves to be restrained, flogged, pinched, etc. but there is no D/s dynamic in the scene when I play with her. Conversely, my wife is submissive and the D/s relationship we have is mostly in play all of the time even though we are not sceneing.
The bottom line really is to find what you like and do that and have fun. There is a growing trend on Fet to use the description Kinkster. I think this is a great way to say, “None of these labels fit me, but I enjoy this stuff.”
So, I guess the short answer is, No, you do not have to be submissive to enjoy being flogged.