I don't think that vagueness is at all a satisfactory way to describe the usage of would in any of the examples that the excerpt quoted.
Here is a much better definition of would meaning OPINION - used to express an opinion in a polite way without being forceful. Here are the examples that my link quotes:
I would think we need to speak to the headteacher about this first.
Here you are politely suggesting that we need to speak to the head teacher.
It's not what we would have expected from a professional service
Here you are politely saying that you expected better work from a professional- for example, an architect, accountant, builder or decorator.
If you find would used before words like say, think, expect, then one possible interpretation is that it is a polite expression of opinion- however many have other interpretations. If you are not sure, check out the would link above for other meanings. Here are some examples that definitely are polite opinions:
Even amidst the unpleasant reality of my divorce, I would expect somebody in the church to understand that there can surely be a note of justice and peaceful resolution in this process. Devotions for the Divorcing
I would think an emigration level at least as large as the 1985-1990 figure (130,000) would be more appropriate. Revised estimates and projections of international migration
I would expect from him enough work that I wouldn't feel guilty about asking him. Don't expect applause
I would say this, that the one thing you cannot do is to act not-acting. Holding and Interpretation
Note that we can make a polite opinion more formal by replacing I with one:
One would expect people to remember the past and imagine the future. Livelihood and Resistance
The example that you asked about:
I would have thought it a proper job for the Army to fight in the rebellion.
The speaker is politely expressing an opinion "It is the proper job of the army to fight in the rebellion". The view is expressed in the past "would have thought", which suggests that actual events differ from what the speaker expected. I don't know the context, but it seems likely that the army has not taken any action to deal with the rebellion.