Thank you for the replies. I think the basic thought of the scorekeeper was that if you score a FC for a batter, it means that there must be an E or out made, because someone should have been out, and no one was. Since a fielder isn't responsible for not catching a bounced throw, it must have been on the 2nd baseman. Thank you for confirming what I believed - that the above is not true. A throw made from the knees 10 feet behind the pitchers mound is not ordinary effort. I think that part of the issue was calling the run an earned run - SK kid is the pitcher. (In fairness and transparency in reporting, my kid is the 2B, and this was his 2nd error in over 100 plays this season, and he wasn't happy about it. In the scheme of things, not a big deal - just wanted to be sure I wasn't mistaken about the need for an E or out since I do a lot of scorekeeping and find the whole process fascinating.)
PWPW - In answer to your earlier question - if the runner from 3rd had just beaten the throw home, I definitely still give the batter a FC, not a hit.