I don’t doubt your sincerity. But I think your legal analysis is wrong.
The correct standard here is not strict scrutiny, it is intermediate scrutiny. This is a much more permissive standard that applies to all commercial speech. And it allows restrictions on what one can say, in order to prevent deceptive practices like those I described.
The Supreme Court described their approach to commercial speech in 1980 (my emphasis):
At the outset, we must determine whether the expression is protected by the First Amendment. For commercial speech to come within that provision, it at least must concern lawful activity and not be misleading.
The Illinois law bans deceptive speech by certain companies trying to gain clients, and therefore it does not violate the First Amendment.