Even if it takes longer than 3 years you can just not leave the US until you get the GC.
The legal explanation for TN visas is the following: An intent to immigrate in the future that is in no way connected to the proposed immediate trip need not in itself result in a finding that the immediate trip is not temporary. Repeated renewal of a TN visa that leads to extended stay in the United States, may still be temporary, as long as there is no immediate intent to immigrate. (9 FAM 402.17-7)
TL;DR what matters is present intent at the time of application and every time you enter the US. It is not illegal if your intent changes after being here for a while.
An intent to immigrate in the future that is in no way connected to the proposed immediate trip.
How I think a reasonable officer would read this: If you are coming in for TN and they ask you about you intent if you say after the next 3 years I am planning on moving back to Canada but in 10 years I want to try to immigrate, that shouldn't disqualify you for the immediate trip.
The trip is the duration of your current stay, not just the entry.
uh, unless you get AP/EAD then aren't you in violation and isn't that reason enough to reject your green card (hell, isn't it reason enough to arrest you and start deportation proceedings?)