It's easy and lazy to throw bombs like, "regurgitating obvious far right soundbites." Yes, I presume those people were a minority, but does that mean they should be stomped into the oblivion of one dimensional shrieking?
The problem is that even though there were likely a minority like that, some play them up as a majority or a significant minority when they usually weren't. And it coincides with antebellum slaveowner propaganda about content slaves who didn't want to be free because they were happy to be slaves. I highly recommend William Freehling's
The Road to Disunion, specifically the first volume, since he spends multiple chapters delving into the mechanics of Southern society, its proposed ideal state and what we know of its real states (since black slaves were mostly illiterate and left few memoirs for us to get their side).
Can you define the two? One could argue the entire united states is historical, we are hundreds of years old after all. Many places try to balance the need to preserve the site with the fact its still in use in some fashion. There is plenty of places where its historical, cultural, and economical. Which factor should we give precedence for?
Also I find a problem with the logic that its cultural, lets assume for the sake of argument you are right. That these monuments are still reinforcing the racial system, then how would they get removed in the first place? If the city or area was still "racist" then they are not going to vote to remove those symbols of racism. And if they are not and do so, then its clear the monuments have failed in their duty and are not actually influencing anything and are harmless on a cultural front, and only "exist" on the historical/economic side of the argument.
Battle sites would fall under historical. Monuments to the men who fought there can explore more of why the battle was fought, the motivations in the leaders and their armies. While there is potential for inappropriate glorification, it is offset by the educational opportunity among other factors.
Museums, likewise, are meant to be educational. They're monuments to the past. Therefore they also fall under historical, although they also have clear cultural intentions as well.
Cemeteries are for the internment of the dead and the preservation of the memory they existed. Can still be educational and is inherently historic, plus it is hallowed ground and cultural taboos against the desecration of resting sites should be sustained.
Statues in grand plazas, however, or public parks meant for recreation, or in front of important government buildings (executive branch offices, capital buildings, courthouses, etc.), are meant for memorialization. They can potentially have historic value, but their primary purpose is to tie the monument to the present state. To glorify the person or persons being memorialized as a standard for the people today to admire and aspire to.
To use the Forrest issue, while my initial reaction was to support his re-internment elsewhere, I would be satisfied if his memorial and such in Memphis made it clear he was being memorialized for turning away from his misdeeds (and crimes) of the past and trying to heal racial divisions. We shouldn't whitewash (heh) the misdeeds of historical figures, especially if they were considered wrong even for their times, but we should also be ready to acknowledge when they tried to improve, to change. Forrest didn't go as far as he could (reportedly he hemmed and hawed before Congress on the KKK, having left the organization but not cooperating with identifying remaining members) but he still went farther than others, and he tried to do good, and that should be recognized too.
The problem I have is that while I agree history shouldn't be erased, the Lost Cause still has a grip on people (witness Texas' decisions regarding the curriculum on the ACW), and these kinds of monuments in public places were made, in whole or in part, to glorify the Lost Cause and preach it to the populace. It is a great historical lie and it must be confronted, not pardoned in the mistaken belief that removing its fruits is "forgetting history".