For many problems, the algorithm with the best asymptotic complexity has a very large constant factor that is hidden by big O notation. This occurs in matrix multiplication, integer multiplication (specifically, the recent O(n log n) integer multiplication algorithm of Harvey and van der Hoeven), low-depth sorting networks and finding graph minors, to make a few. Such algorithms are sometimes called Galactic algorithms.
Note that for other algorithms, such as general sorting and integer addition, algorithms are known with optimal asymptotic complexity and small constant factors.
What research has been done in separating the former algorithms from the latter algorithms, from a theoretical perspective?
I am aware that hidden constants are often omitted to hide the distinction between different models of computation. However, I am confident that under a wide variety of different models, these Galactic algorithms will be slower than asymptotically worse algorithms for inputs of size one billion, for instance. The distinction is not subtle, in some cases. Has it been made rigorous?
For instance, one could invent a very simple model of computation, such as a von Neumann machine with a very simple ISA, and then implement the algorithms and bound their running times with explicit constants. Has this been done for a variety of algorithms?