I think strategy can exist on the micro level (ie, trying to pull off one shot, or figure out how to make a birdie on any one hole) without keeping score for all 18. I think the architecture can also be experienced in this manner.
However, for me I get the most enjoyment out of trying to put together a full 18 hole round. I think approaching a round in this way provides more opportunities to be strategic. For example, there may be a stretch of holes designed in a way that don't play to your strengths and another that you feel comfortable being more aggressive on. Managing a round to me is fun and trying to decide whether to carry the water on the fifteenth hole versus laying up is more interesting to me in the context of a full round rather than on just playing the 15th hole independently.
Coming down the stretch and trying to figure out a way to break 80 (or 90, or 100, or par, or whatever score is meaningful to you) is what I consider fun and a way to experience the architecture in a meaningful way.