Why can't we mimic a dog's ability to smell COVID?
We can actually detect some diseases via smell, and the term to search for is olfaction.  The general problem is known as breath analysis.
However, the research into olfaction and machine learning is rather new (perhaps even surprisingly new).  As Lötsch et al. point out, little research (prior to the very recent research) on olfaction and machine learning has been performed, with a few exceptions:

Quantifying olfactory perception: mapping olfactory perception space by using multidimensional scaling and self-organizing maps, Mamlouk et al., Neurocomputing, 2003.
Relationships between molecular structure and perceived odor quality of ligands for a human olfactory receptor, Sanz et al., Chem Senses, 2008.
Diagnosis and Classification of 17 Diseases from 1404 Subjects via Pattern Analysis of Exhaled Molecules, Nakhleh et al., ACS Nano, 2017.
And the one mentioned above, Machine Learning in Human Olfactory Research, Lötsch et al, Chemical senses, 2019.

I don't know whether the problem in general is harder, but as you are touching  on in your question, the problem is much harder from a hardware perspective.  Where imaging only needs a simple camera, and hearing only need a simple microphone, to detect smell you need a so-called
as chromatography–mass spectrometry instrument. As the Wikipedia article mentions:

Breath gas analysis consists of the analysis of volatile organic compounds, for example in blood alcohol testing, and various analytical methods can be applied.

Here are some pointers from popular science that should assist you in getting into the literature:

Scientists Invent An AI That Can Smell 17 Diseases From Your Breath, Including Cancers

Innovative AI Breath Analyzer Diagnoses Diseases by “Smell”

AI is acquiring a sense of smell that can detect illnesses in human breath