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So you're right that CTC is acting weird. I've noticed this few years ago too on different board, but I've always suspected asychronous mode (I've added 32k RTC Xtal on Arduino Mega).

However Fast PWM mode with OCR2A as a TOP value seems to be working:

class TimerTwo
{
  public: //methods
    void initialize();
    void attachInterrupt(void (*isr)());
    void detachInterrupt();
    void (*isrCallback)() = nullptr;
};

TimerTwo Timer2;

ISR(TIMER2_COMPA_vect) { //ISR on Compare Match A
  Timer2.isrCallback();
}

void TimerTwo::initialize() {
  cli();
 
  OCR2A = 124;                        // = (16*10^6) / (2000*64) - 1 (must be <256)
  TCCR2A = _BV(WGM21) | _BV(WGM20);   // turn on fast PWM mode with top in OCR2A
  TCCR2B = _BV(WGM22) | _BV(CS22);    // Set CS22 bit for 64 prescaler
  TCNT2  = 0;                         // initialize counter value to 0

  sei();
}

void TimerTwo::attachInterrupt(void (*isr)()) {
  isrCallback = isr;
  TIMSK2 |= _BV(OCIE2A);
}

void TimerTwo::detachInterrupt() {
  TIMSK2 &= ~_BV(OCIE2A);
  isrCallback = nullptr;
}

volatile uint32_t counter = 0;
void isrHandler() {
  ++counter;
}

void setup() {
  Serial.begin(115200);
  Timer2.initialize();
  Timer2.attachInterrupt(&isrHandler);
}

void loop() {
    static uint32_t counter_old = 0;
    Serial.println(counter - counter_old);
    counter_old = counter;
    delay(500);
}

EDIT: Ok, so after some digging on this issue I've found that setting CTC mode might cause OCR2A corruption (I guess?) . It seems to be working if you reorder the settings to:

  TCCR2A = _BV(WGM21); // turn on CTC mode with top in OCR2A
  OCR2A  = 124;         // = (16*10^6) / (2000*64) - 1 (must be <256)
  TCCR2B = _BV(CS22);  // Set CS22 bit for 64 prescaler
  TCNT2  = 0;          //initialize counter value to 0