Does s a landlord have to issue a Form 1099-NEC to a self-employed contractor who provided services totalling more than $600.00 during the year?

I depends on whether they are / wish to be considered a trade or business......

If they have profits and wish to qualify for the 199A deduction, acting like a trade or business, would include issuing 1099-Misc and 1099-NEC as required for business activities.

Maybe?  Does the activity rise to the level of a trade or business (hint: are you taking a QBI deduction for it?)  Were the payments made by cash/check or credit card?

The limit is "$600 or more" and not "more than."

You might be thinking of the law that was repealed in April 2011 (10 years ago) just as it was taking effect:

"Another law under the Small Business Jobs Act of 2010 added recipients of rental income from real estate to the list of taxpayers subject to the same information reporting requirements as taxpayers engaged in a trade or business. This law was effective for payments made after December 31, 2010. Rental income recipients making payments of $600 or more to service providers in the course of earning rental income were required to issue a 1099-MISC to the service provider. For payments made after December 31, 2011, the same expanded reporting requirements that applied for business taxpayers (issuing 1099s to stores for the purchase of supplies) would also apply to rental activities."

What if they are reporting the activity on Schedule E?

Maybe it depends, but the IRS instructions for Schedule E wouldn't lead you to believe that. 

" In general, if you
paid at least $600 for services performed
by someone who is not your employee
(nonemployee compensation), you must
file Form 1099-NEC"