The formula for marginal likelihood is the following:
$ p(D | m) = \int P(D | \theta)p(\theta | m)d\theta $
But if I try to simplify the right-hand-side, how would I prove this equality
$ = \int \frac{p(D, \theta)}{p(\theta)}\frac{p(\theta, m)}{p(m)}d\theta $
... and so on? I can't seem to simplify it. I can't just "remove" $ \theta $ here right like you would do if there was only one expression? As in this isn't the same as:
$ P(D)\frac{P(m)}{P(m)} = P(D)? $
So the marginal likelihood is more of a definition than a result, what you do always have from basic probability theory is the marginalisation $$ p(D|m)=\int p(D, \theta | m)d\theta, \tag{1} $$ so there is an assumption that $p(D | \theta, m) = p(D | \theta)$ - this is a hierarchical modelling set up.
Since we know $(1)$ my comment is just that as quickly as possible you want to go $$ p(D|\theta)p(\theta|m) = p(D, \theta |m), $$ so instead of an expansion like you have considered you are actually wanting to condense everything to a joint density.