While this probably doesn't fully answer your question, at least it might be able to serve as a start.
I think the biggest conceptual difficulty here is regarding what exactly the heart of the Lucas critique is, and why it was so influential. Before he published his seminal 1976 work a large body of macroeconomists were primarily engaged in attempting to investigate policy analysis by estimating fundamental parameters of a model (at the time, since Keynsian analysis was in vogue, it generally focused on things like the marginal propensity to consume or save, or estimating consumption functions for use in policy counterfactuals, etc.). How this was done was by using aggregated historical economic data to try and back out the parameters of interest.
The Lucas Critique points out that this effort cannot be guaranteed to be internally consistent. That is, all of the observed, aggregate historical data being used was actually the output of a complex system. When considering how a change in policy might impact welfare, the economist cannot assume that these parameters will remain unchanged.
Critically, this is true independent of whether the model is using rational expectations. No matter the macro modeling assumptions, the internal optimization that the agents are "doing" inside the model must be grounded in actual fundamentals of economic theory. That is to say, there's no theoretical reason why one consumption function should be used by an individual over any other (for example). Historical data, then, can only give insights into what the optimal decision making behavior was at the time it was observed.
There's actually a pretty good example on the Wikipedia page for the Lucas Critique in the last section (behavior surrounding Fort Knox). Rational expectations would suggest that individuals are maximizing some, known objective function for whether they should try and rob the vault or not. Given that we don't see many attempts, we can then say that, given individuals are following rational expectations and are maximizing some consistent utility function, in general, the perceived costs outweigh the benefits. The Lucas Critique, on the other hand, says that we cannot use the fact that we don't observe many attempts at robbery now to suggest that Fort Knox spend less on security. While RE theory says people aren't robbing because the costs outweigh the benefits, the Lucas Critique says we cannot use the fact that we don't observe many arobbery attempts at this level of security to say anything about how robberies will change if we change security levels. All we know is that, in this particular set of circumstances, we observe some equilibrium behavior (few robberies). To make policy statements, we have to base our models on some fundamental theory- one of the most prominent being a form of RE. But ultimately, whether we believe RE holds or not, the Lucas Critique still points out that we must base our predictions in some theoretical foundation, and not only use aggregated outcome data.
Sorry that was kind of long winded and sloppy- I'll try to tidy it up later if it would help!
Is it fair to say that the LC is a criticism of keynesian models because they didn't include expectations of any kind rather than just not rational ones?
Broadly, that's exactly it. Keynseian models at the time needed lots of aggregated parameters (again, like marginal propensities, etc.) to come up with conclusions on how different policy changes, or economic shocks, might impact the economy (or industry, or group of people, etc.). The problem is, Keynsian analysis (again, at the time) requires these parameters without any theoretical context. Why should the marginal propensity to consume be X? What would cause that to change? These questions require the microfoundations that we've seen spring up in the macro field- models that attempt to examine what determines important macro parameters, and how changes to those underlying causes will change the parameters of interest. Not only did Keynsian models not include expectations (specifically forward looking variables), they tended not to include any theoretical basis for the parameters they estimated.
And, as long as the included expectations can be connected to either A) the policy rule or B) the underlying stochastic process ( if there is no specific policy rule sometimes they seem to do this), the model is then internally consistent.
While these are certainly two good microfoundations that an economist can use to build a macro model, model consistency (again, using the broadest possible definition) can rely on an even weaker basis. For example, you could have a model of the macro labor market where labor supply is completely divorced from any sort of rational or forward looking behavior at all. You could base your model on something like:
Every period, all agents wake up and flip a fair coin. If the coin is heads, they go to work. If it's tails, they go back to sleep until the next period.
Of course, this premise is completely divorced from reality- coin flipping has nothing to do with the decision of whether to go to work. But it would at least provide a clear theoretical basis (even if absurd) for why the labor market behaves the way it does. Instead of just observing "last period, the labor market participation rate was 60%, thus, we should take that as some underlying parameter" (like the classical Keynsian methodology might), you'd say "I see last period there must have been an inordinate number of coins that landed 'heads.'" Your theory would give you a specific reason for believing the participation rate was 60% (the proportion of heads), a prediction for what it would be next period (50%, since that's the expected number of heads you'd expect to see next period), and a reason ** why ** it should be 50% (again, because everyone's flipping coins).
Moreover, using the coin-flip model of labor markets, you could consider how a policy change should affect behaviour. How would a doubling of wages change turnout? In this case, we'd predict it wouldn't have any impact (still doesn't impact the coin probabilities). And while that might seem absurd, at least you can very clearly explain why there will be no change, using some economic reasoning. The old Keynsian models couldn't provide these types of explanations, other than "because the last time wages doubled, X happened to participation." While that might be nice to know, it cannot be considered a useful answer, since there might have been any number of other things causing the wage change to happen the last time, that might not be at play now (and which might have been even more important in driving the participation changes).
Rational expectations, then, is just one useful, generalizable, easily applicable mechanism for how to bring microfoundations into these macro models.
Again, hope that helps!! Let me know if I can clear anything up!!