So first things first, whether or not Stand Your Ground is in play, the burden of proof is always on the State to prove any crime did happen and any defense does not.
Another thing that I think you confused in your question is it seems apparent that you think Stand Your Ground is Self Defense. This is not true.
In the United States, self-defense is always a legal right for a victim of a potential crime, regardless of if your state has Stand Your Ground or Duty to Flee laws. Self-Defense typically can include justifiable homicide as you are not privy to the intent of the bad actor.
Under Duty To Flee laws, you cannot claim self defense if you could reasonably get away from a criminal action safely... if given the choice between fight or flight, you must flee the scene.
Stand Your Ground contradicts this and says that if you are in a public place and a criminal is trying to make you a victim, you have every right to defend yourself without any duty to remove yourself from the situation first... basically at this point, you can make either choice and not worry about losing justifiable Homicide.
Making a criminal arrest of a Stand Your Ground claimant at the seen is not necessarily required. While the claim may be disputed, in the case of firearms, using an illegally owned weapon is typically ground for arrest regardless... (probably not in cases where the illegal gun was introduced to the scene by the dead criminal... and the victim picked it up in a scuffle... though this requires some measure of sorting out). Legal Fire Arms are very well documented and the fire arm in question will be confiscated as evidence. If it is found that it was not a justifiable homicide, the person in question is probably at the address tied to the gun.
Now, again, Stand Your Ground only applies to steps needed for Self-Defense, it is not self-defense itself. Self-Defense authorizes only the amount of force needed to safely resolve the situation, up to and including leathal force, but it does not require you to kill the perpetrator in every instance it is invoked. For example, if merely pointing a gun at a perpetrator is enough to stop the crime, you do not get to pull the trigger. That flips it back into homicide. Similarly, if I pull my gun and the guy advances anyway, I may fire and if the guy is on the ground and out cold (thus, no longer a threat), I don't get to walk up, and put a second bullet between his eyes, execution style. This too is murder.
As a bit of anecdotal evidence, when I was living in Florida, I worked for a man who just recently purchased a firearm for self-defense (in the home only) and he said that when he was filling out paperwork with the police, the cop looking over his paperwork said, "Now remember, if you have to use that, shoot to kill. It's less paperwork for us."
Now, I wasn't there when to cop said it, I don't know what his tone was. I took it as the cop being a little funny, but maybe a little inappropriate. I cannot speak to how much that is indicitive of FL Police culture. It was hearsay on my part... I just thouht it was funny and... demonstrates the attitude towards self-defense. Essentially, by the time cops arrive at the scene, they HARD PART is over... they merely have to collect evidence and take witness statements. If the shooter is cooperating and his story checks out, it will look very bad if they detain a crime victim who defended himself. It's just bad PR. Ultimately, his job is to collect all evidence, not determine if the case should go to trial. As I mentioned, the gun was legally owned in the specific case, and more than likely the CCTV tape is collected, but not yet viewed. Hindsight may be 20/20 but at the time, I do not think it's fair to say that the cop knew this might not be such a clear cut case.
In such cases, the cop may not make an arrest because there is not any crime that he can charge the man with and he is cooperating. And keep in mind that in the heat of the moment for the shooter, he may not even realize he did something that might break his self-defense case.
Cops can detain a person claiming Stand Your Ground for just about any legitimate reason, even suspicion of homicide that the detainee will claim is self-defense.