https://en.wikipedia.org/wiki/Abductive_reasoningModern cosmology got kicked off in the 1700s by observations of "nebulae" that showed many of them were collections of stars, and in particular that some were much much much larger and more distant than others. Just before WW I the absorption and emission line structures of the spiral ones were discovered to be strikingly similar except the smaller (in angle) dimmer (in apparent magnitude) ones were squashed into the red.
Just after WW I is when spiral nebulae were identified as anything remotely like our modern understanding of spiral galaxies. 1920: https://en.wikipedia.org/wiki/Great_Debate_(astronomy) [poor Shapeley, so bright and so so wrong on this point] about five years after a working theory of post-Newtonian gravitation was even available, and almost exactly two years after General Relativity aced its first observational test in the solar system. Up to that point even the greatest names in astronomy (even Einstein) believed everything in the sky was within or in a close (~ kiloparsecs) orbit around the Milky Way.
Towards the end of WW II and just after radio astronomy became important, particularly the study of the 21cm hydrogen gas line, which was clearer than the red-squashed lines of visible light passed through prisms, and in particular different limbs of galaxies had different redshifts, proving rotation. Some three decades later, the 21cm redshift difference between the inner and outer parts of a number of galaxies showed that there is non-Newtonian gravitation obviously at work in large galaxies. (Also coincidentally around that time, the cosmic microwave background was discovered, but it was some years before the small anisotropies in it could be studied -- BOOMERaNG and COBE in particular to start with).
The evidence in all these cases arrived in advance of vague ideas, and forced the hunt for tractable explanations for the evidence in its totality, rather than as individual stand-alone pieces. One of the biggest pieces to fit in is of course the highly successful standard model of particle physics, which also was driven by evidence arriving kinda by surprise somewhat concurrently with surprise evidence from cosmological observations.
The result is the "concordance cosmology", \Lambda-CDM, which concords with all the available data (well, or rather it's updated as new data shows up from various observatories and experiments). It's certainly subject to speculation: what's the microscopic description of dark matter? is the cosmological constant actually uniform everywhere in spacetime? is there a non-cosmological-constant term required to match new data for the Hubble flow? These are pretty big questions, but they're forced on us by the in-your-face obviousness of the metric expansion of space, and the peculiar motions of galaxies within clusters, and the outer parts of galaxies around the inner parts. Also, what's going on in "the dark ages"? We have an obvious gap between the surface of last scattering and the first starlight, but the details of the observed first starlight and the cosmic microwave background don't interpolate as well as one would naively expect. And we have so many exabytes of data about the latter (and a lot of data about early galaxies too) that vague ideas die quick deaths: they don't even get a chance to generate new "falsifiable hypotheses", they are generally born inconsistent with some existing data.
The "trick" is abduction: trying to reason out a simple-enough-to-be-useful explanation that fits the known data.
Of course, you can get away with wild speculation and vague ideas in areas where there is little to no data at present. Anything before the electroweak decoupling is anyone's guess, as is anything much more than a trillion years in the future, or far outside the Hubble volume.