lme4, failure to converge with a range of optimisers, trust the fitted model anyway?

One of the problems is that you have a relatively high random effects
variance. A standard deviation of the intercept of 3 is a huge amount, it
means that there is massive variation in the random effect value needed to
model each cluster, to the point that some clusters will be all zeros and
some will be all ones. In this situation the assumption of approximate
normality of the likelihood around the nodes which is required for using
Laplace's method is very far from met.

Thanks for your advice, I really appreciate it!

I tried nAGQ=5, but met with:

## Error: nAGQ > 1 is only available for models with a single, scalar
random-effects term

As you point out, some clusters will be all zeros (and some will be
all ones). While my data is on the individual level,

a) the variable I'm mainly interested in, KilledPerMillion5Log, varies
only at the country level, &

b) I currently have no variables in the model that vary at the
individual level

So, perhaps I could aggregate the individual level data to the cluster
level, and do without the random term for cluster? I mean, calculate
the proportions of yes in each cluster and use that as the dependent
variable.

This would, I assume, require that each cluster was given a weight
that corresponded to the number of individuals in it - or I would not
be able to say anything about probabilities at the individual level,
right?