This is typical of the kind of compromises we have to make from time to time. Every trade is unique in the way it unfolds.
I do not often move my SL especially if the trade is working clearly as anticipated. But sometimes movements are very sluggish and takes all day to get there and creates a sense that the market could reverse at any time soon......In these situations if the price is anywhere near my target then I will either close out some of the position or raise the stop level if there is a newer sensible location for it.
But I think the key to this is in your statement:
If you are no longer interested in the trade then it means that the initial reasoning behind your expectation for the trade has evaporated in some way, in which case, I would simply close it out with the current profit and look for the next trade. There is no need to painfully sit with a current trade just to see "what will eventually happen"
But one has to be a little wary. As you rightly say too aggressive adjustments to stoploss levels can easily turn an ultimately winning trade into a loss. Sometimes the price can take a long time before it finally reacts in the way we have anticipated and it is easy, and extremely frustrating, to close a trade just before it takes off without us!
In principle, unless there is a sensible new level to move our stop to, or the price has just missed our target and seems to be in danger of reversing, or we feel the trade is somehow "wrong", then it may be better to leave the settings as originally planned.
There is no right or wrong answer here. And this is just my own take on it!