"Is your harassment complaint legitimate enough to me that I should get up and do something about it" is 100% completely the wrong answer.
It's less about viewing the complaint as legitimate or not and far more about having the necessary information to actually do something genuinely effective. Just issuing warnings to people who may not have the slightest idea of why they are being warned does absolutely nothing to fix the problem. Zero tolerance policies that automatically assume that the accusation is correct will ultimately cause as much harm as good because you are not actually giving the type of information needed for others to actually learn what needs to change. You are simply telling them that someone doesn't like them and that doesn't provide any kind of help to someone who might have been doing it completely unintentionally. Everyone keeps assuming that the harrassment is legitimate and that the harrsser fully understood what effect they were having on the accuser. In this day and age where people seem increasingly convinced that they are free to interpret things however they feel they want to and have no responsibility to attempt to understand what was actually intended, I cannot assume that the complaint is for an intentional act. I have never said don't take it seriously or to brush off the complaint. I am simply saying that if you are going to create and enforce a policy, it needs to be geniunely fair to everyone. Your hypothetical solution doesn't actually resolve anything when increasingly the chances of the person being accused not actually understanding why they are getting the warning and therefore have no ability to reduce their chances of getting another one are quite high.
If a person has so big of a problem that they cannot effectively articulate it sufficiently for the convention staff to give an effective warning that lets the person know at the mimimum what triggered the warning, that person does not need to be in that situation without some kind of help or supervision, because they cannot expect convention staff to intervene on their behalf effectively if they (or a friend) cannot effectively translate at least the nub of the problem from their head to someone else's. I get that people need to be careful about how they ask the necessary questions and convention staff should absolutely get at least some training on effective conflict resolution, but the people complaining about this behavior have to learn that they are as much part of the solution (or problem as the case may be) as the convention staff or those being accused of harassment. Simply saying "I'm being harassed" is not enough; if the person is doing it unintentionally and that's all they get told, they can't fix it. If the person is doing it intentionally, there may be grounds for immediate removal or even greater powers getting involved immediately. Simply assuming that all claims are true and doing absolutely zero investigation may solve the immediate issue someone was having, but it does so by potentially creating bad feelings toward the topic and deprives everyone else of the deeper level of information needed to genuinely understand the problem.
Trying to prevent these people from suffering anything in the short term actually helps create the environment in which they will continue to be at greater risk of feeling pain repeatedly in the future. That is the biggest reason I thoroughly despise this line of thinking. People who have that kind of deep problem never learn how to actually deal with it sufficiently to be comfortable around others and others never learn enough about why it's a problem to be able to adapt their actions to be proactive in combatting it rather than relying solely on a convention staffer coming up to them and telling them they did something someone didn't like. People who see these moments solely from the perspective of the person making the claim fail to understand half the problem; it is not just about enforcing an anti-harassment policy, though that is certainly part of it. It is possible, and our society is getting really close to this, to get so focused on individuals and their problems that we forget to each ourselves and each other the necessary skills and empathies needed to function in public without some kind of outside help (in this case, convention staff or a game store manager). People who are so afraid of being hurt that they can't even tell the proper authority why they were hurt need to learn that functioning in society requires a bit of effort on their part as well a bit of effort on everyone else's part; dealing with negative emotions effectively (or at least effectively enough to get along in public) is simply something that everyone has to learn because it's something that everyone has to deal with at some point or another.
Our society has become so terrified of dealing with anything negative that a lot of people have concept of how to deal with it when it becomes unavoidable, and that's where much of the core problem lies. Past generations never had these issues to the degree we see them today because they had ways of dealing with and addressing them that our current society lacks because we have become terrified of rejection, risk, and pain in general. Proceeding with caution does not mean completely shielding the accuser from any pain or backlash that may come from any particular difficulty. If someone came up to me and made a complaint based solely on the phrase "I'm being harassed," I cannnot help them. Even if I give a formal warning to whoever they want me to like you suggest, I have not helped them. They still lack the ability to be truly comfortable in that public space because they have no idea if the message was fully understood, and everyone around them lacks the knowledge about what triggered the incident to make things easier for them to come to terms with the problem. All I have done is put myself in a situation where I am ultimately the bad guy to both sides, because chances are very good that unless I automatically assume that every complaint is valid and end up throwing out half the attendees, I am being insensitive to the needs of those who rightly or wrongly believe themselves to be victims, and everyone else is left being very annoyed with me that I didn't even make an attempt at due process for those being accused.
I guess for me it comes down to the fact that it very much feels like a lot of people feel that they have the right to interpret everything they see or hear however they see fit, but at the same time expect everyone else to fully respect their own personal intentions and thought when trying to interpret their actions and words. They cannot have it both ways. Either everyone is free to interpret things however they please and everybody better grow a real thick skin real quick or there must be some expectation of not just assuming the worst and that others are not bound to act based upon you assuming the worst. The legitimacy of the claim is usually not the actual reason a manager or staff member would want to carry out at least some kind of investigation, and people who are so sensitive that they are incapable of distinguishing between a legitimate investigation and criticism should not put themselves in a situation that they might need to. Whether that means bringing a friend to help cover the details they cannot easily manage or accepting that they simply need to remove themselves from that public space until they can manage those details sufficiently for those in charge to do more than a token slap on the wrist of someone who may not even understand why their wrist is being slapped.
This may mean that some people simply won't be able to go certain conventions or gaming stores until they figure out how to deal with their problems enough to get along in public; if that is the case, so be it, the world will not stop spinning because one individual doesn't get everything they want when they want it. Just because I have a deep personal problem (and everyone does at some point or another) does not mean that I am justified in believing that I can go anywhere I want and force complete strangers to deal with the fact that I have a problem. Any resolution of that problem begins with me; others can help, but in order to do so, they have to have sufficient knowledge to do so, and if I am not comfortable providing it, it is unreasonable for me to expect them to reach a resolution that I am comfortable with. The ability of others to help is entirely dependent on my own ability to articulate the problem effectively; the vast majority of people don't read minds and I for one like it that way. The expectation that all I have to say is I'm being harassed is sufficient to solve the problem just pushes the problem down the road without solving anything and it stands a good chance of making things worse.