Why, and how, are barometric altimeters accurate enough for approach and landing?
A difference of 1 hPa results in an error of 27ft in the altimeter. To have the altimeter showing 3000ft when the aircraft is actually at sea level, the altimeter must be off by 111hPa. Standard air-pressure at sea level is 1013 hPa (that's converted to 29.92 inHg for you Americans).

For comparison, the pressure in Hurricane Katrina reached as low as 902 hPa. So if you were flying into a hurricane but left the altimeter setting at 1013 (which is used as the standard setting at cruising altitude), you could theoretically pop out of clouds at what you thought was 3000ft but instead hit the ground. 

Ignoring the fact that nobody tries to land in a hurricane, this mistake is prevented by the airfield pressure being reported by ATC, in the ATIS, and even an alert from the aircraft if it is advanced enough. 


  While on approach, you enter a localised low-pressure cell that the airport’s AWOS, being at the airport rather than back along the approach path, doesn’t know about. Seeing your altitude apparently ballooning up, you idle the throttles and push the nose over to correct the altitude excursion; unfortunately, since you were never actually too high in the first place, this action instead sends you below the glidepath, and straight into the ground.


Pockets of low pressure as you describe just don't exist to that degree. Hurricanes take hundreds of kilometres to reach their low point. And anyway, this is why airliners have a requirement for the approach to be stabilized at a specific point -  typically 1000ft above the aerodrome level. Any significant deviation from the glideslope below this point mandates a go-around. 


  During a go-around, you set TOGA power on the throttles and pull back on the yoke; as the elevators are quicker to respond than the engines, the aircraft pitches up before it can accelerate much, resulting in the aircraft’s angle of attack temporarily increasing. Ram air enters the static ports, the altimeter says you’re lower than you actually are, and, in an attempt to avoid hitting the ground, you pull the yoke as far back as it’ll go, stalling the aircraft.


Altimeters are calibrated with multiple static ports and designed so AoA changes do not noticeably affect the altimeter. For the rest of this example, I mean I suppose a pilot could just forget the basics of flying but it's unlikely. Pilots know how to select the correct pitch and power to guarantee maximum climbing performance. 

You didn't mention that altimeters are also affected by temperature, and in cold air they will read higher than reality.  There is no correction for this, and so some approaches will have limitations in very cold weather. EDIT: See John K's comment below.