This might depend a bit on the black set-up.
One reason to play Qe1 instead of Rf1 could be that the rook will often be needed on the f file later to support the attack with f4...
Another example where Qe1 can make sense in the KIA is to support a knight on h4 when playing f4 as in positions like the following (which is fairly random) where white can recapture on f4 with the g pawn.
r1bq1rk1/pp1n1ppp/2nb4/2p1p3/3pPP1N/3P2P1/PPPN2BP/R1B1QRK1 w - - 1 1
One problem with Qe1 besides the one you mention is that the pawn on c2 is unprotected and could be attacked by black with Nb4. Playing a3 to prevent Nb4 is also not something white usually wants as it creates targets for black's queenside pawn storm attack.