As a Canadian, what’s the best way to advertise to potential employers my eligibility for the TN visa, and emphasize how much less hassle it is than H1-B visa sponsorship?
Somehow as soon as possible in the application/recruitment process, let the potential employer know that although "sponsorship" is required, the process can be quick, cheap, and easy.
SUPER easy: https://www.uscis.gov/working-united-states/temporary-worker...
You do not need a lawyer and the letter of employment can be done by the HR department of your employer, just be sure to include everything they ask for. I did it all by myself and got my company to reimburse me the fee after I came in to work.
For sure sell yourself with a TN Visa, there's NO sponsorship required, just disclose the fee payment. I'm a Mexican who did the whole process by myself, no lawyers from the company required, and even though it's easier for Canadians since you just go to the border with the requirements and pay a super small fee, you get right in and can start working the next day after you come into the states. Assuming your profession is in the TN NAFTA list, you can get to work within a week with a TN Visa.