Quotes from Uplinker:
(1)
"The White cross is NOT a flight director, NOR is it any sort of aircraft attitude indicator. The white cross shows the sidestick position ONLY."
Indeed. Describing the white cross as a "sidestick target", as in the title of this thread, is misleading. I'm not sure where the OP got the expression from, actually. It's 11 years since I flew the a/c, but my original A320 FCOM, dated Feb 1988, describes them (one per PFD) as follows:
"On ground, side stick position indications appear white on CPT and FO PFD."
(2)
"I guess this potential confusion, (although frankly it beggars belief), might be why Airbuses now remove the white cross very soon after rotation."
Yes, but you could omit the "now". As one of the first line pilots to fly the A320 (March 1988), I can assure you it was ever thus.