When you are not paying people what you are really willing to pay them, eventually they'll learn about this, and hell will break loose. You either have to change the way you pay, or be OK with losing people. Counteroffers get you the worst of both worlds, and on top of that, they are wonderful sources of lawsuits in the long run, as the people that won't complain are typically women and minorities.
The one counteroffer that make sense is to go and say 'look, now we realized that we are underpaying, but we want to keep you all, so we are doing an immediate salary review'. You still look like someone that isn't on the ball, but at least you show you are trying to have equitable pay.
But being paid below market rate also sucks. By making a counter-offer and having it accepted, you're now setting a precedent. All of that person's peers will now see that the way to get a raise at this company is to constantly be interviewing and threatening to leave. Is that the environment you're trying to foster?
Trying to get a raise with another offer is a bluff. You have to live with that fact that you might be called on that bluff.
I think you meant to say "without another offer". When you indeed have an offer, it's not a bluff. Whether you get a counter or not, the outcome for you is you win. Now, if you DON'T have another offer, but try to get a raise by claiming to have another offer--well, that's an advanced tactic and very risky.
>the way to get a raise at this company is to constantly be interviewing and threatening to leave
If your goal is to get a raise and keep your job, coming in with another offer is a bluff. The way to get a raise at my company isn't to constantly be interviewing because we won't counter offer everyone and we generally won't offer repeated counters.
Where is this mythical "market rate" that I as an employee or employer can constantly get access to?
Every employee has their own market rate and that rate is determined by BATNA negotiations. If I'm hiring someone, I don't have market rate for them in mind which I try to pay under.
As a manager at a small company, you barely have more info on market rates than employees do. Sometimes savvy employees even have more info than I do (they can go out an interview to assess the market, while I can't interview people just to see what offer they'd accept).
Of course, there are also cases of flagrant underpayment (ex. paying <$100k for an engineer in SV) which should drive you away automatically.
I've seen most of the options in action. The workplace can be an evil place.