How long can a Canadian stay in the US as a visitor after the TN status 60 days grace period?

Does this mean I can leave the US and return to the US with the TN status for 60 days, or do I have to stay in the US for the 60 days in order to maintain the TN status?

Every time you leave the US, your TN status ends. You can only be readmitted in TN status if you qualify for TN status at the time of admission. So no, you can't leave the US and re-enter in TN status if you're in the grace period. The actual regulation is 8 CFR 214.1(l)(2):

(2) An alien admitted or otherwise provided status in E-1, E-2, E-3, H-1B, H-1B1, L-1, O-1 or TN classification and his or her dependents shall not be considered to have failed to maintain nonimmigrant status solely on the basis of a cessation of the employment on which the alien's classification was based, for up to 60 consecutive days or until the end of the authorized validity period, whichever is shorter, once during each authorized validity period. DHS may eliminate or shorten this 60-day period as a matter of discretion. Unless otherwise authorized under 8 CFR 274a.12, the alien may not work during such a period.

Note the use of the word "consecutive."

Various articles online say that I can stay in the US as a Canadian visitor up to 183 days within 1 year.

The 183-day threshold is a product of US tax law. A Canadian who spends more than 183 days in the US will become a resident alien under US tax law, which makes their entire worldwide income subject to US taxation. You are probably already a resident alien for US income tax purposes, in which case you don't have to worry about that.

As far as immigration law is concerned, your admission as a visitor will normally be for six months per visit. You will be in B-2 status unless you mention that you will be attending job interviews, in which case you should be in B-1 status. Six months is of course close to 183 days, but it's not the same thing. It could be 181 days, or 184 days, for example, if you enter on January first of a non-leap-year, or on March first, respectively.

In fact, the date by which you must leave the US is not normally the result of an automatic calculation from your date of entry. Normally it is determined by the immigration officer who admits you to the US and recorded in your I-94 record; it may also be written on an admission stamp in your passport.

You should check the I-94 record at https://i94.cbp.dhs.gov, because the US is in the habit of issuing "multiple-entry" I-94s, meaning that on any subsequent entry you could be readmitted for the balance of a previous period of admission. You should keep track of this so you don't end up inadvertently overstaying. Whenever you re-enter in visitor status you should consider whether you are planning to stay beyond the last "until" date you were given, and, if so, point that out to the officer.

Needless to say, that could cause problems for you, since Canadians are not supposed to live indefinitely in the US in visitor status. You might get some leeway by saying explicitly that you're planning to remain in visitor status for a finite period of time before seeking employment as a TN professional. But then again, saying that might work against you if you encounter the wrong immigration officer, who could easily twist that around into a disqualification for visitor status. The best way to ensure your ability to remain in the US will be to find another job that qualifies you for TN status as soon as possible.

Is it safe to say that I can stay in the US for another 6 months after changing my status to visitor without too much scrutiny from the border agent?

Once you're in, you probably won't face much scrutiny from much of anyone until the next time you try to enter. Your problem when you try to enter will probably be credibly explaining how you plan to support yourself for six months in the US without working illegally, or credibly claiming that you plan to spend a shorter time in the US if in fact that is what you're doing.