Why did the women go to the tomb: to anoint the body, or to see the tomb?

These are not alternative reasons. To anoint the body the women had to go to the tomb, and the first thing they would do there was look at it; the anointing was what they meant to do once inside. Mark shows both in one breath: they bought spices to anoint him, and on the way they asked one another who would roll the stone away (Mark 16:1–3). Matthew’s verb theōrēsai means to look at or gaze on, and he has already shown the two Marys “sitting over against the sepulchre” on the Friday (Matthew 27:61), watching where Jesus was laid. On the Sunday they come to look again. Barnes suggests they wanted to see whether the tomb was as they had left it, and Gill that they wanted to see whether they could get in to anoint the body.

Matthew may leave the spices out because his account turns on the sealed and guarded tomb and on what the women saw and heard there, not on a plan that events overtook. John follows Mary Magdalene and tells what she found, not why she came. A writer who gives one purpose does not deny another, and a writer who gives none denies nothing.