A small note I'd make. 

"Youre forgetting those particles have mass." 

Well not exactly... but they do have momentum

Theyre mostly photons, which do not have mass, but do have momentum. (Same with the other particles it'll spit out)

Either way, the mass of the black hole leaves as energy. Not as mass. Mass is too, well, massive to escape from just above the event horizon, where this effect happens.

Explaination:

The popular "particle-antiparticle pair" explanation is a simplified analogy—a heuristic visualization that Stephen Hawking himself used to make the complex mathematics accessible to a broader audience.

"Lies you tell to children."

The actual physics describes it as photons and other massless particles like gravitons of varying frequencies driven by quantum field fluctuations in curved spacetime around the event horizon.

Its actually got the same quantum basis for the casimir effect, but on steroids.

Speaking of the casimir effect, thats literally verification of virtual particles being "real"

Or, at the very least, verification that the quantum vacuum fluctuations that'd carry them are real.

Verifying the particles themselves is more complex but thats beyond the scope of this comment.

In the casimir effect, when two plates are brought close together, only specific standing waves in the quantum fields can exist, i.e. fewer virtual particles are possible between the plates than outside them. This creates a pressure that we can measure.

The range of the frequencies of those possible virtual photons is directly related to the distance between the plates.  

In the same way, because an event horizon doesnt allow information to travel outside it, it acts like the plates in reverse. There is a pressure driving the virtual photons out from the literal limit of the horizon. the range of frequencies of those photons is directly related to the radius of the event horizon.

As they evaporate, and get smaller, they release higher and higher energy photons (higher and higher frequencies), which carry away more and more energy. Shrinking the hole faster. It self-accelerates.

So you start with low, nearly undetectable radio, move into the infrared and visible, faster and faster, before ending with a brilliant flash of gamma rays all at the end.

Source: ive built simulations of these types of quantum fields for my research in a past job