… but the Style would need a mirror in the Compile Styles in the Compile Format Designer.
I’m not sure what you mean by that, as any style not addressed by compile settings should implement how it looks in the editor to the best of its ability. One does not have to do twice the work, in other words, to style some text. You style it, and it compiles. It would be an awful design if every style you made while writing completely vanished when compiling.
The only reason to go in and fiddle with compile settings is if you don’t want the bold text to be bold. Then you add a rule, and then you set the formatting you want it to be.
When I compile, the italicized text is coming through into the compiled ebook (Kindle format) as are the tables. But NONE of the bold text is in bold.
To the OP, there are probably too many variables involved here to give you a simple answer. For example, you might be using a font that doesn’t have a normal bold variant but is using Black or Semibold, and that is confusing things. It may be the ebook itself is perfectly fine, but what you’re reviewing it with is not. Those are just two possibilities.
What works best is to make a little sample project of what you are trying to do, show how it doesn’t work, then use File ▸ Back Up ▸ Back Up To... with the zip compression option enabled, and drag the .zip file into your response here to attach it.
bold_text.zip (138.4 KB)
That is what I have done here, to get things started. Please open this, compile it, and see if it works for you. As you should see when examining the output directly, the only real fault here is the output spraying so much redundant bold on top of everything it’s a miracle you don’t have bold. 
<h1 id="doc1" class="title">Bold Text Test</h1>
<p class="ps1">First test: <strong><span style="font-weight:5600;">This is done using direct formatting</span></strong>.</p>
<p class="ps1">Second test: <strong><span class="strong-emphasis">This is done using a proper named style</span></strong>.</p>
And yes, in case you were wondering, the CSS file has more bold for the strong-emphasis class.
.strong-emphasis { font-size: 1.00rem; font-weight: bold; }
If you are getting the same HTML, but still are not seeing bold when compiling that test, then the problem is indeed with the ebook reader. If that does work though, then I’d start looking at differences between your project and this sample.
