The question is, why?
The first observation is important in its own right. Going back to my extremely limited experiencing teaching macro to freshman, I would run through the Keynesian cross first without G (of course its in their textbooks with G in a fiscal policy section so it came in pretty quickly). But the first cut was nice to do that way precisely because I'm one of those sticklers on Nick's first observation. The important contribution is the theory of how the economy works, not the policy view. Plus in the General Theory he doesn't have G in his initial Keynesian cross analysis (in fact I'm not sure if it ever comes in explicitly). So call it my history of economic thought rebellion against modern textbooks.
Nick also gives two important reasons for why Keynes thinks that the division of expenditures between consumption and investment might not always be optimal - neither are particularly commonplace in casual internet conversation. If you're speaking to a Keynesian critic they'll usually cite "animal spirits" and say that Keynes basically handwaved at the whole question. Somewhat more sophisticated conversations will bring in Nick's "macro reason" - the problem of how Say's Law breaks down in a monetary economy. I have almost never seen anyone bring up the "micro reason". Nick doesn't cite to the General Theory but this comes from Chapter 16 in what I think is a too much neglected passage:
"AN act of individual saving means — so to speak — a decision not to have dinner to-day. But it does not necessitate a decision to have dinner or to buy a pair of boots a week hence or a year hence or to consume any specified thing at any specified date. Thus it depresses the business of preparing to-day’s dinner without stimulating the business of making ready for some future act of consumption. It is not a substitution of future consumption-demand for present consumption-demand, — it is a net diminution of such demand. Moreover, the expectation of future consumption is so largely based on current experience of present consumption that a reduction in the latter is likely to depress the former, with the result that the act of saving will not merely depress the price of consumption-goods and leave the marginal efficiency of existing capital unaffected, but may actually tend to depress the latter also. In this event it may reduce present investment-demand as well as present consumption-demand.This is exactly why when people tell you that Keynesians think that investment and capital are homogeneous they have no !&#$% idea what they are talking about and should not be taken seriously on the matter. It is one of the dumbest criticisms out there.
If saving consisted not merely in abstaining from present consumption but in placing simultaneously a specific order for future consumption, the effect might indeed be different. For in that case the expectation of some future yield from investment would be improved, and the resources released from preparing for present consumption could be turned over to preparing for the future consumption. Not that they necessarily would be, even in this case, on a scale equal to the amount of resources released; since the desired interval of delay might require a method of production so inconveniently “roundabout” as to have an efficiency well below the current rate of interest, with the result that the favourable effect on employment of the forward order for consumption would eventuate not at once but at some subsequent date, so that the immediate effect of the saving would still be adverse to employment. In any case, however, an individual decision to save does not, in actual fact, involve the placing of any specific forward order for consumption, but merely the cancellation of a present order. Thus, since the expectation of consumption is the only raison d'être of employment, there should be nothing paradoxical in the conclusion that a diminished propensity to consume has cet. par. a depressing effect on employment.
The trouble arises, therefore, because the act of saving implies, not a substitution for present consumption of some specific additional consumption which requires for its preparation just as much immediate economic activity as would have been required by present consumption equal in value to the sum saved, but a desire for “wealth” as such, that is for a potentiality of consuming an unspecified article at an unspecified time. The absurd, though almost universal, idea that an act of individual saving is just as good for effective demand as an act of individual consumption, has been fostered by the fallacy, much more specious than the conclusion derived from it, that an increased desire to hold wealth, being much the same thing as an increased desire to hold investments, must, by increasing the demand for investments, provide a stimulus to their production; so that current investment is promoted by individual saving to the same extent as present consumption is diminished."
Monetary policy is of course also correlated with Keynesianism, but it's correlated with other ideas too, because monetary policy moves the investment/consumption mix. I think fiscal policy is correlated with Keynesianism for a different reason: because fiscal policy is considered, like investment, to be exogenous w.r.t. income. You can think of the exogeneity of investment in the Keynesian cross sense that investment is not a function of income but consumption is (this should be reminding you of the Landsburg discussion and some bad counter-examples that people came up with - specifically the problems Malcolm outlined in his comment that I shared as a post), or you can think of the exogeniety of investment in more neoclassical terms as a function of technical parameters in the production function, depreciation, time preference, etc.
However you choose to think of it, fiscal policy is an exogenous lever that gets income back on track - and when income is back on track then private preferences can get the distribution of expenditures back on track.
As a side note I think that a good appreciation of externalities means that there's lots of stuff we should be confident in the government spending on anyway (so why not make it counter-cyclical?). But this is more of a Pigovian point than a Keynesian point.