In short, the special theory of relativity predicts that photons do not have mass simply because they travel at the speed of light.
Based on outdated ideas, from a quick look.
There was a massive gap in the speed of light <> infinite mass concept when Einstein came up with it; it's all based in classical physics which in turn is based on objects (masses) only being moved or accelerated by external means.
As I noted in #14, relativistic mass is added by the action of external acceleration, adding energy and equivalent mass; projectile or ballistic physics. That now appears to be a generally accepted explanation.
A photon is never accelerated, it is created with it's natural velocity.
Escape velocity is a good example or the classical physics omissions; nothing being able to leave a gravity well without reaching that velocity - a purely ballistic physics concept.
It does not apply to any vehicle with continuous propulsion that can exceed the attraction of gravity for a sufficient time. There is nothing yet that could work like that in earth gravity, but possibly such as an ion drive for low gravity moons or asteroids etc.
And the never-talked-about one - If relativistic mass increase is due to external energy input, a self-contained spacecraft can never experience any mass increase at any velocity, as there is no external energy input.
Remember that the scientific community has phenomenal inertia against acceptance of change to existing standards, even when the existing explanations of a concept or phenomena are blatantly incorrect - probably the craziest example being that it took near enough a hundred years for the explanation of how aircraft fly to be updated from it purely relying on the greater upper curvature of the wings meaning a longer, faster, airflow path than the lower surface. That of course meant inverted flight is impossible, which has been known to be very obviously incorrect since not all that long after aeroplanes were invented. It was stupidity to the point that some scientists in the early or mid 20th century claimed that inverted aircraft or those with symmetrical profile wings or blades were not technically "flying".