Note that you have good answers - and this isn't an answer per se but too long for a comment.
However this rule is just stupid.
It is antiquated and has carried over because of MLB - at all lower levels I am sure they would change this and some leagues have rules overruling this (see later). MLB hasn't changed it because:
MLB doesn't like to make changes that effected past stats. Which is just dumb because this literally hardly ever happens at the MLB level.
Since it doesn't happen at the MLB level, no one gives a crap about it. The first time someone won or lost a batting title based on a baserunning error ESPN would do a documentary and MLB would happily change the rule.
I did scorekeeping for a few high school teams my sons played on. I told the coaches from the get go that I counted hits as hits, rules be damned. Doing otherwise doesn't let the stats reflect what is going on, on the field. Just like when giving out errors in baseball or assists in basketball it is about obeying the spirit of the rules.
Men on first and second. Batter lines a one hopper to the left and the shortstop dives and catches the ball. Notices that runner from first is already at second. Runner that was on second was caught daydreaming and thought there was no one on first so decided not to try to advance. He would have been safe by a mile but is out on force. Why in the world would the batter have a "FC" on his score card. My score card shows "1B" for batter and baserunning error and out for the kid forced out at third.
Man on first. Batter hits a hard line drive to left. Runner was moving from the start. Leftfielder was playing in and picks up the ball clean off one hop and fires to second base for the out. That is a "FC". The batter caused the out. Highly unlucky but that is his out.
Man on first. Batter hits a bloop well over third baseman's head. Runner decides to tag when he should have been in the middle of first and second. Leftfielder doesn't get to it but fires to second and beats runner by a step. Again this is a hit "1B" and a baserunning error.
Now if someone is superbly slow... We had one player like that. He literally took 7-8 seconds on a good day to get to second with a lead off (and he wasn't fat so it fascinating). I charged him with 4 baserunning errors that year. It simply isn't measuring the hitter when he keeps getting forced out at second from the outfield. I kid you not one play the ball was batted between center and left and dumb "Chad" (yes he wasn't bright either) takes like 4 steps towards second then shifted his momentum back to first. The centerfielder had to backtrack big time so the ball didn't pass him. The batter probably had a 95%+ chance at a double and surely would have went for two. Chad got nailed.
In my opinion if you did this right you would set the right attribute to the right stat. For instance a batter who hits a one hopper to the right fielder and trips on his way to first, that is really a 1B and baserunning error. The at bat resulted in a "hit". The baserunning resulted in an out. So in conclusion the rule is absurd. If scorekeepers get to give a third basemen an error after he lays out for a ball but then bobbles it, well they can handle these obvious baserunning issues a few times a year. How you score things is up to your league and if your league doesn't have rules it is up to you.