[Reply about Blanchard-Yaari OLG models and Gertler (1999) paper. I will post separately later about Carvalho, Ferrero & Nechio (2016).]
Short answer: No, VFI Toolkit cannot solve Blanchard-Yaari OLG models like in Gertler (1999).
Long Explanation: Blanchard-Yaari OLG models are NOT heterogeneous agent incomplete market models which is what VFI Toolkit is built to solve. While they can technically be considered OLG models as in any time period there are two generations, workers and retirees, they depend on some very specific mathematical assumptions to largely eliminate any heterogeneity. One way to think about it is that Blanchard-Yaari OLG models are to OLG models what TANK models are to HANK models (Two-agent New Keynesian models, Heterogeneous agent New Keynesian models).
I will explain with a bunch of references to Gertler (1999) [this is such a well written piece that it would be silly for me to not use it 
]. As you note in your post Gertler (1999) has a utility maximization problem for the worker (written as a recursive problem) in his equation (2.3). He then takes the first order conditions of this to get a consumption euler equation, his eqn (2.11). He can then aggregate across consumers to get an aggregate consumption rule, his eqn (2.15). He does something analogous for retirees (2.3, 2.5, 2.14). He can then combine these two to get aggregate consumption C_t in eqn (2.17). What is key is that he uses only this last equation in the definition of competitive equilibrium (Definition 1 on pg 74 refers to eqn 2.17, but not of the others I just mentioned that were used to derive 2.17). Hence while he ‘microfounds’ equation (2.17) in the end he does not use the microfoundations to solve the model.
So in the end the model to be solved is just what is given in Definition 1 on pg 74. If you write out all the equations (Definition 1 just refers to them by number) you will see that they form a system of difference equations. This system of difference equations is thus all that the model to be solved consists of. In this sense it is the same complexity of model as the neoclassical growth model (a.k.a. Solow-Swan growth model) which is also a system of difference equations.
If you are familiar with New Keynesian models, this is analogous to Chapter 3 of Jordi Gali’s book, where a household problem is used to derive the basic New Keynesian model, but the final model is then just a system of stochastic difference equations. Systems of stochastic difference equations can then be quickly and easily solved in Dynare and we can forget all about the microfoundations from which it was derived. In Gertler (1999) once we derive the system of difference equations we can just solve these and forget all about the microfoundations from which it was derived.
So these Blanchard-Yaari OLG models, like Gertler (1999), are really more like the Solow-Swan model that standard OLG models. Part of the giveaway is that as Gertler (1999) mentions on pg 74, “it is possible to express all the endogenous variables as functions of the two predetermined variables, K_t and Lambda_t”. To put it another way, the state space of the model is just the capital holdings of ‘all workers’ and ‘all retirees’. The distribution of capital across ages, or across different workers or different retirees is irrelevant. By contrast in any standard OLG model the distribution of assets across ages is going to be an important part of solving the model. In a model with idiosyncratic shocks (and incomplete markets) the distribution of assets becomes a much larger and more complex object and is a key part of solving the model.
A quick explanation of why Blanchard-Yaari OLG models are able to kill off most of the complexity of the model. While the original household problem (Gerlter (1999) equation 2.3) might look complex, there are three very important assumptions. The first is stochastic ageing, which essentially means age itself is irrelevant, there are just workers and retirees [for example this means there is no such thing as ‘nearing retirement’]. The second is annuities in assets of retirees, which means that the risk of death does not introduce any differences in agents. The third is the weird preferences (following Farmer (1995)) which means that the workers are risk neutral (to the risk of retirement). Normally any idiosyncratic risk, like retirement or death in this model, would generate heterogeneity. But by making markets complete for the death risk (this is what annuities are doing), and making agents not care about the risk of retirement (because they are risk neutral), the framework is eliminating heterogeneity. In some sense this is strongly related to a point that used to be often made about Real Business Cycle models: adding idiosyncratic shocks to the basic RBC model doesn’t change anything as long as we have complete markets (the point remains true, you just rarely hear it now that we can just solve the incomplete market heterogeneous agent models anyway).
A final comment on Gertler (1999). This is a seriously nice model because the analytical tractability means you can show a lot of the main concepts of standard OLG models just directly in the equations. This analytical tractability does however come at the expense of quantitative performance.
Summing up. Solving Gertler (1999) involves solving a system of difference equations (those listed in his Definition 1 on page 74, or it’s extension on page in Definition 2 on pg 83). So it is simpler than a RANK model, which is a system of stochastic difference equations, and much simpler than heterogeneous agent incomplete market models (and so cannot be solved with VFI Toolkit). In principle a system of difference equations is easy to solve, although I am not personally aware of software/toolkit for doing so. The model is of the same ‘class’ as the neoclassical growth model.
But I find it almost impossible. Now you, as the author of the toolkit, confirm my guess .