Akinbo,
All cosmology becomes nonsense if photons don't experience time. Astronomy is all based on spectroscopy and delta lambda, which gives the absorption history of each particular 'photon' of light (or WHATEVER wavelength) analysed. Each is different, most can be matched well to it's journey. Reverting to old myth and legend fudged to cover the anomalies will only continue to confound us.
THE BUS. You got close but stumbled at the last minute and fell off! Forget the LT for a moment the runner doesn't 'stumble' when he jumps on the bus, he is 'accelerated' into the new rest frame. Just think about that alone for a while to embed it. (And remember 'deceleration' is also properly 'acceleration').
Now that is simply the 'Galilean Transformation'; an 'acceleration'. All very simple and entirely logical yes? (and light ON the bus always propagates at c wrt the bus, as we know).
Ok. Now on to the LT, which is also simple but a DIFFERENT and additional effect only becoming a factor at high relative speeds.
For this the bus is rapidly approaching a line of say 5 runners 1 metre apart who each jump on in turn. The 1 metre gaps then clearly reduce. That is 'Doppler blue shift' (shorter wavelength). OK so far?
Bit when the bus is going VERY fast there's clearly a problem as wavelength can only reduce so much! The runners will be squashed together as they jump on the bus. The last two may even just bounce off! Are you following? There is a minimum possible wavelength (gamma). If the transform requires a shorter wavelength then the light simply can't penetrate! The excess energy is emitted as heat (i.e. check out the nose plasma bow shock of the shuttle on re-entry, where even radio waves can't penetrate! That's why the plasma density at gamma is called 'optical breakdown mode'.
That is the LT, an additional factor added to the GT as a 'power curve' when relative speed approaches c. Now if you did the homework I set it would be clear as a bell and I wouldn't have to repeat all this over and over! (lol)
Imagine the electrons trying to oscillate but not having the elbow room. If they can't oscillate then light cant get through. The propagation speed of light ('between' particles that is) is then set by a 'rev limiter' the min electron wavelength gamma (electron refractive index is n=1, same as the 'vacuum').
I recommend you should really read that lot again to ensure it gets assimilated! If you can't find the logic of any parts or would like to see the mass of consistent evidence do just ask. (Is that Tom I hear having a confounded fit!?)
Best wishes
Peter