Not really. Most AC software is proprietary and owned by for-profit companies, and these very rarely make software open-sourced. Whether it's more secure or not is not a factor.
Not sure if you're saying that for-profit companies don't make open source software frequently, or that they specifically don't make open sourced anti-cheats, because the former is certainly false.
While there is ways to data mine, reverse engineer, and/or circumvent close source software and this code is not publicly audited, I think we'd both agree that it is probably better than just giving cheat developers the keys to your game.
Having the source code is more like having the blueprints to a lock than the keys. If the lock is a bad lock, then having the blueprints will make it easier for an attacker to bypass it, but if it is a good lock, it doesn't matter whether the attacker has the blueprints or not. Hiding the implentation behind layers of obfuscation is called security through obscurity and is generally considered a bad practice as it allows security flaws to stay present for longer than if the implementation could openly be scrutinised by everyone. Not only malicious hackers can look at the source code and find flaws in the code, anyone can.
As for the open-source discussion. Companies defintely don't frequently develop open-sourced software compared to how often they develop proprietary software. Almost all software I have installed on my computer is proprietary software. The reason for this is simply to make more money and prevent theft of code, not to enhance the security.