I am mystified by the scientists who keep on working and making propaganda for already falsified theories (or tell me where are the trillions of black holes surrounding galaxies, hiding in "temporal pockets" à la dr Who?).
Despite you encourage "open minds" you seem to believe that there´s only one theory compatible with the principles of scale relativity, yours. Then the paradigm is dead. To clarify, I´m not talking here about adjustable parameters, all parameters are univoquely defined once the equations "work out".
Singularities are a by product of the Platonic attitude of taking flawed theories too seriously _ or even perfect theories: a theory is not reality. Einstein himself NEVER believed in a primordial singularity (neither you, I see) nor in "singularities at the heart of extremely collapsed stars". Missing (or dark) matter might be a by product of ignoring electromagnetism on cosmic scales so gravitation is the only actor. Gravity and electromagnetism are relevant on all domains, I see you agree on this.
Two random ideas:
normally, space is considered infinite but, what if space turns out to be unbounded but finite, a 3-sphere, for instance? Then we can not find arbitrarily large scales or objects (self similar or not), the same way we can not fit any size on Earth surface.
In part inspired by the R - 1/R duality of string theory, it may be that scales do not go indefinitely up and down but they close, ergo, largest scales and tiniest scales match. For instance, subquantum and metagalactic realms are one and the same. Not infinite nesting of matter but not upper or lower limits either.
Regards
PS: whenever you find a singularity, call me. By the way, merry christmas and happy new year.