In the 1930's yes this would be true. But we have many pieces of evidence today. These include:
* The temperature and anisotropy angular power spectrum of the cosmic microwave background
* The observation of galaxy evolution with redshift
* The observation of elemental abundances that agrees with Big Bang Nucleosynthesis
* The Lyman alpha forest (in Quasar spectra) evolution with redshift and the Gunn-Peterson test
* Time delays in gravitational lens systems (which give us a measure of the Hubble constant as a funciton of redshift)
* Observation of the Baryon Acoustic Oscillation peak seen in galaxy redshift surveys (evidence of the hot origin of the early universe)
* The observation of the evolution of structure in the Universe with redshift
* Agreement between the amplitude of structure in the CMB and the Universe today (When evolved forward with simple theory)
There are many more but that should keep you busy. It is not on a whim that the ideas of modern cosmology have been developed, they have been developed precisely because of the overwhelming evidence. Alternative ideas were reasonable when there was less evidence, for instance Fred Hoyle's steady state model, however new evidence ruled these out leaving the current model as the main contender at present, but that could all change with new evidence.