Restrictions on free speech do exist, although the specifics in Canada, the US, and any other country someone might be reading from will differ and if we're going to refer to them, we'll have to use the US ones because this is a US case. However, it is probably useful to understand that they aren't written as such. The law doesn't say "you have free speech except in the following cases which can be changed by the legislature". Restrictions come when some other harm, such as that experienced by people who can't enter a hospital when they need to, overrides it. In many cases, this difference may look academic, because the result is the same, so the causes can't be that important.
There are two cases where that stops looking as unimportant. The first is in a court of law. If a court is going to decide that restrictions on the content or conduct of speech are necessary, they have to provide a reason that is compliant with other laws. In this case, that reason is "national security", which is now to be debated. The generic argument that you started with, that it isn't restricted if you could go somewhere else, is much broader than that and not supported by any law, which leads us into the second case where your version and the version currently in use look very different: in a world where there are active abuses.
There are always people who think that their free speech should have no limits whatsoever who think their inability to block a hospital door if they want to without being arrested is a violation of their rights. Those people are wrong. Most others understand why that is not allowed and wouldn't want to do that anyway. When free speech is eroded, the restrictions that are placed almost never say the right has been eliminated. They just put more and more restrictions on how you can exercise your right. For instance, when there were protests in the US about civil rights, the states that sought to block them never did that by passing a law saying that those protests were not allowed. They passed laws saying those protests were definitely allowed, because those people have the same speech rights as anyone else, but coincidentally you couldn't protest in the places where those people were likely to start doing it. They would then arrest people who tried. That was an illegal restriction, struck down several times by courts, that takes the form that you defended in your original argument. That is why your argument is incorrect from a legal perspective and undesirable if you could write it into law and why, even though it is, TikTok may well lose their case because it doesn't protect them. They will be protected or unprotected on their free speech claims by the court's interpretation of the national security value of a ban.
TikTok had some better, in my mind, legal arguments. This law appears to me to be a bill of attainder, a law that simply declares a certain entity is criminal without criminalizing any action they do. Those are not allowed. If I were a lawyer, I would have tried that. Their lawyers did and a court decided it didn't apply. As I am not a lawyer, let alone a constitutional lawyer, I don't know enough to know if the court had a good reason or not to decide that. I thought that was more likely to succeed than any free speech argument, and my opinion on that has not changed.