For something to be iconic it has to be functionally present in the environment. Whilst yes the white tailed deer does range into the amazon it both doesnt have a large range there and it isnt a large proportion of its own range.See, I don't really get that argument. whether or not a species is "iconic" is super subjective. I consider superb lyrebirds to be an icon of Australia when barely anyone's heard of them. I don't really get why species are only seen as mono-continental (the greater flamingo and grey wolf are worse offenders, granted), it's just an arbitrary limitation to limits creativity and thinking. Maybe we should normalise the idea of white-tailed deer as Amazonian species, even if it seems weird at first. Doesn't hurt anybody and it enables more freedom.
The white tailed deer is a south american animals and an amazonian animal but it is definitely not iconic to the amazon only ranging in the outskirts of the forest and in areas where there is not alot of people. It could be considered an iconic south american animals but that is fairly nuanced with its status as an iconic north american animal getting in the way and the fact it only ranges in the north west even if its very common there. Add that to the large majority of internationally known areas of south america being outside white tail deer habitat and it doesnt generally get perceived as iconic.
Grey wolves as a whole have never been seen as mono continental just that the generic one we have is HEAVILY inspired by a north american subspecies both in name, scenery and style so to some people its only usable in that context. The greater flamingo isnt mono continental either and I havent heard anyone say otherwise except maybe that there isnt enough other animals to build a proper Mediterranean section and doesnt really fit with our asian animals.