Yes most of us have this idea, somehow. It's strange isn't it, how ignorance overpowers the mind. "Yes, yes, death comes, but not now, but not me." "Yes of course I know, I know. Death can happen that way. But no, not to me!" It's amazing how ignorance can deceive us isn't it? That's the power of Mara. There you go. It comes to deceive us.
"Is complete and honest acceptance of death for the enlightened?" I don't know. We certainly have to get some moments where we do believe we're going to die in there, before we really have that motivation, what we call Samvega, a sense of urgency in our life. This is why, in the Satipattana Sutta, the Buddha is always asking us to contemplate when we see a dead body externally, reflect that "also my own body is of the same nature, such it will become." You have to do it over and over again to break through the power of Mara and the power of ignorance in the mind, the ignorance that believes somehow that we're not going to get old, we're not going to get sick and we're not going to die, that we're somehow immortal.
"Is complete and honest acceptance of death for the enlightened?" Actually I don't believe the enlightened accepted death. That's my opinion. From what I understand about enlightened people according to the scriptures, they're free from birth and death. So, they didn't even want to continue to be born and die and born and die and born and die all the time. Rather perhaps they saw, "Is this really what I want to be in, this cycle of birth and death?" Perhaps they came to a realization that if they continue to accept that they're going to be in this cycle, then they will have to accept that they're going to die and die and die. But perhaps they came to a realization that this is Dukkha, this complete cycling in birth and death and birth and death is Dukkha, and they wanted to rise above it.
Accepting that we will die certainly is a motivator to want to develop ourselves and free ourselves from this Dukkha. The more moments that we can get in there, to accept that we're going to die, the more Samvega or sense of urgency that we will have to develop ourselves to be able to free ourselves from Dukkha may arise. But the power of ignorance is very strong as we see. We can be reflecting about death and we still have this feeling, "no, it's not going to happen to me." However the more we do reflect about death, then the moment may come where we may be thinking we're dying, and this is where fear can arise and then we think, "Well, I'm going to die anyway, I knew it could come this way, what am I going to think about?"