Are there any disadvantages to HRP conjugated PRIMARY antibodies
You can compare directly coupled antibodies to the classical indirect primary-secondary systems:

Directly coupled antibodies

Advantages:


quicker workflow as only one antibody is used
eliminates the chance of a cross-reacting secondary antibody (possibly less background)
possibility to use differently labelled antibodies on the same blot


Disadvantages:


lower sensitivity because the signal is not amplified
only few labels are available
directly labelled primary antibodies tend to be expensive and you need relatively high amounts of them
the labelling can damage or reduce the immunoreactivity of the
antibody


Indirect Primary-Secondary Antibody systems

Advantages:


the secondary antibody amplifies the signal (which is especially
interesting for weakly expressed proteins)
differently labelled secondary antibodies can be used on the same
primary
primary antibodies from different species can be used simultaneously
one secondary antibody can be used with different primary antibodies
(for example for the protein of interest and the loading control)
the labelling reaction does not affect the primary antibody


Disadvantages:


cross-reactivity of the secondary antibody can lead to non-specific
bands
the method takes longer, as more steps in the process are necessary


For me the most important point is the higher sensitivity of the indirect method, for comparision see these images (from Life technologies):