Okay, this is a situation that we see a lot here on the Boards. There is a reason why Support cannot tell you the details of what 3rd party program they found. It's not that you talked to a "bot." You did speak to a live Support Specialist, but Riot's policy is that any time a malicious third party program (scripts, hacks, etc) is discovered on an account, they do not reveal the information about when it was discovered, or how they discovered to the owner of the account. This is because the hackers/scripters can then simply alter the hacks and scripts to make them unable to be detected in that manner, which makes it much harder for Riot.
That being said, let's look more into your specific case. You said your account was "legally boosted with xp boosts." What do you mean by that? Do you mean that YOU bought a bunch of XP boosts and just played a ton of ARAM matches to level it up to 30? Or did you have someone leveling the account for you? Many of the people who level accounts use 3rd party programs that are forbidden, so if you had someone else doing it, that could be what triggered it. Not to mention, account sharing is a huge no-no. But if you leveled it yourself, then we don't need to worry about that.
Aside from that there are other possibilities of what could have triggered the ban. You can check this thread for a list of some of the common 3rd party programs that Riot has responded about whether they are allowed or not. Many times, people don't even realize that some of the programs they have on their computer can interact with League.
So, looking over that thread, I would have to ask you some questions. Did you buy the account or create it from scratch? Do you use any kind of macro? Do you have a gaming keyboard/mouse with additional macro functions? Do you run any custom skin programs? Do you run any sort of timing program? Do you run any sort of video capture program? Do you play other games on your computer, if so, do you have any programs installed that are there to modify them in any way? Is your computer your own private computer or does anyone else have access to it (friends, family, etc)? If anyone else has access to your computer, do they also play League? Have you ever let anyone else play on your League account? Have you ever logged into your League account from another computer, either at a friend's house, family member's house, public library or internet cafe?
If you answered yes to any of those questions, it will give us a place to start. I know it is very frustrating when you try to contact Player Support and cannot get any direct answers about why you were banned for a 3rd party program. It's very, very tempting to believe that it has to be an unfair and completely wrong ban, when they won't give you the proof that they have. But these bans are usually quite well researched prior to being delivered. It's just a matter of us trying to figure out what the trigger was. And we'll try to do everything we can to help you find it, so that you can avoid having it happen again.