A very thrown-together and hopefully not misleading illustration
All of this is quite confused, and ironically gives you not just a non-Keynesian, but a quite anti-Keynesian view of the economy. I like to think of the Keynesian approach to investment and consumption in pretty basic Econ 101 terms - I like to say that Keynesians "want to move along the consumption demand curve, but they want to actually shift the investment demand curve". It doesn't exactly translate over from microeconomics, but here's an illustration of what I mean:
Let's start with the left panel. Here we just have simple loanable funds market. Supply of loanable funds is constant, but demand for loanable funds shifts to the left because firms increase their preference for liquidity and decrease their preference for new investments. The same loanable funds are available. So without getting into any assumptions about reduced consumer demand, we now have aggregate demand shifting down purely due to the reduction of investment demanded at market prices from I' to I''. Now - should we assume that consumption demand changes? Maybe a little - cautionary saving is certainly plausible on the part of households for the same reason that liquidity preference is in play for firms. But there are good reasons not to change consumption demand relationships that much. Keynes thought that the marginal propensity to consume out of income was a psychological law - certainly informed by time preferences, but not something that shifted around a lot. This is what we see in the data - consumption volatility does not play a very big role in the business cycle compared to investment volatility. Keynes certainly would have been aware of this.
What you see on the Keynesian cross panel on the right is the shift in income resulting from this reduction in investment demand from C+I' to C+I''. Because the diagram assumes some autonomous consumption (i.e. - there is some consumption out of savings even when income is zero), the share of income that is consumed is increased somewhat as income is reduced (this stands to reason if investment is more volatile than consumption). I've added a horizontal line that I think may be useful for understanding how consumption shifts in a Keynesian framework - it's the dashed line I call the "fully depressed investment level of consumption" (FDILC). This is the level of consumption when investment is so depressed that there is zero investment. The first vertical line extending up from the FDILC notes the additional consumption above FDILC for the C+I'' equilibrium, while the second vertical like notes the additional consumption above FDILC for the C+I' equilibrium. What you'll note is that investment demand recovers, we will see a recovery in consumption too. You will observe improvement in consumption data, without any changes in underlying consumer preferences or confidence. Unlike the investment level I never moved that consumption curve. The change is due entirely to an increase in income - in other words, the existing consumer demand became more "effective" as more income came online to spend.
Implications for consumption, a few links, and humanitarianism
So, what does this say about consumption boosting policies like unemployment insurance? Well to be clear, first and foremost it says they might do something. If you actually can boost the consumption curve you are going to shift out to a higher income level. The problem is, you are remedying a symptom and not the underlying disease. As the FDILC line illustrated, the fluctuations in the consumption level are a result of the changes in the income level. You can bolster that consumption short-fall, to be sure - but you're not doing anything about the underlying cause. If investment demand is still depressed, then as soon as you turn off the unemployment insurance spigot you'll go right back to where you started. Maybe consumer confidence plays a big role in the reason for depressed investment demand. If that's the case, then investment may recover somewhat. But Keynesians usually point to more than just consumer confidence - particularly if there are concerns about a liquidity trap.
This all means that consumption-driven policies are not entirely useless, but in the end they're symptom-treating policies from a Keynesian position. As I've pointed out in the past, you can see this in the General Theory, where Keynes himself ridicules the very idea of unemployment insurance. A real Keynesian perspective is pretty agnostic and skeptical when it comes to unemployment insurance, which I think is illustrated by recent thoughts from Greg Mankiw on the UI extension that you can read here. One of the posts that actually got me thinking about this problem was one by Russ Roberts where he ties concerns about consumption to Keynesianism. The post is simply titled "Keynesian sentence of the day" - and the sentence from the Washington Post that he's refering to is "After two years on the sidelines, American consumers are spending again and raising hopes that they are ready to shoulder the burden of the nation’s economic recovery." What frustrates me about Roberts and many other commenters on his blog is that it seems like they think "anything a liberal ever says about the economy is Keynesianism". The blog is rife with suggestions that Keynesians want to see people consume more and invest and save less. Another example is this video featuring Hiwa Alaghebandian which was recently shared by Don Boudreaux as a video that demonstrates "some of the flaws of Keynesian economics":
The video is riddled with problems (a major one being the unsupported assumption of perfect crowding out), but one of the most obvious problem was the consumption-centric interpretation of Keynesianism. It's a little embarassing, because Hiwa is a senior at my alma mater, the College of William and Mary. Maybe there has been staff turnover, but I know that none of the macro professors that were there when I was there would have taught her this. She has, however, had internships at Cato and AEI. I imagine she picked up this understanding of Keynes there.
So what do we do about consumption-related policies like unemployment insurance extension? Different people are going to split different ways on this, but I'm personally cautiously sympathetic to UI extension. One thing you won't hear me arguing quite as often is that we should think of it as a depression-fighting policy. Because I take the Keynesian view that most of our problems have to do with investment demand, I primarily support UI extension as a humanitarian policy and a structural unemployment policy. The humanitarian justification should be obvious - workers are unemployed through no fault of their own in the midst of a tough job market. They have obligations and they have fulfilling lives to lead. It's simply a humanitarian imperative, apart from any macroeconomic concerns, to provide a cushion for them. But there are other more calculated economic reasons for doing it. Unemployment insurance supports active job search, and that will help keep "the unemployed" from becoming "the unemployable". Long-term unemployment poses serious risks of higher structural unemployment in the future, and UI can help fight that. Ideally, to do this UI would be paired with options for training, job search assistance, and new hire incentives for businesses who hire unemployed workers, or even public employment options. Again, though, there's no argument here about the cyclical benefits of UI.
So are there any cyclical benefits? There are a few. First, as I noted above, shifting the C curve up will improve output, and this may improve investor outlook. UI is also paid out of unemployment insurance trust funds - which are exactly what they sound like: pools of saved payroll tax funds that are invested, but kept relatively liquid to be paid out in the form of unemployment checks. Drawing down on these trust funds can be thought of as making use of relatively idle capital. In other words, we're not just shifting money earned today from the employed to the unemployed: we're shifting money from an idle fund to the unemployed.
Mostly, though, I think of UI and many other consumption-oriented policies as being humanitarian and perhaps structural policies, rather than primarily cyclical or "Keynesian".
This message leaps off the page of the General Theory, too. The General Theory is all about investment, and much less about consumption. As I like to point out, Keynes has (if I remember correctly) about twice as many chapters dedicated to investment as he has dedicated to consumption, and he's famous for advocating the "socialization of investment", not the "socialization of consumption". And when consumption is addressed, it's usually addressed as a stable relationship that investment oscillates around.