Hazel be aware, I am speculating playing with ideas. Hand waving
Hoyles steady state model or something similar, might be more plausible than the Big Bang if for instance the CBR is evidence of ongoing particle creation.
Any way an inflation stage all over the universe, causing baryogenesis, (based on Dark Matter) or MAYBE something similar to Hawking radiation whereby virtual particles are driven apart, becoming baryons(quarks) (No imaginary Dark matter required) which form all the matter in the universe giving off radiation in the process, forming the CBR.
If particle creation is a low level on going process then Hoyles Steady State model or a variation becomes interesting, and appears quite plausible, especially if the radiation pressure caused by the CBR is driving the expansion of the universe. Either way both Big Bang and Hoyles ideas require particle creation, likely both by inflation, both would result in CBR. One happens in a flash of a second, and never happens again. The other happens continuously as a fact of how the universe works.
In the Hot Big Bang instance, all matter and radiation were produced in a smaller region of space, and expanded outwards cooling down as it goes, dark energy drives the expansion of the universe.
Perhaps under continuous particle creation, all matter created in the universe gives of CBR, which would heat space, but due to the expansion MAYBE driven by the CBR space stays cold at around 2.75Kelvin.
Yes, I am aware that you are speculating. And I am riding along for the fun of it. Seems to me that we are all still in the speculative stage of where the beginning - if there ever was a beginning - really begins. Do you remember "I Love a Mystery"? :-)