"Electoral institutions" is a pretty broad term. Looking at the discussion on differences between representative and direct democracy might be useful in this context.
From an empirical social science perspective, there are a myriad of possible theoretical and empirical qualifications to the ability of elections to represent peoples' interests.
Karl Popper was a powerful defender of democracy and the "Open Society". Wittman  does a good job defending democracy against common arguments from economics.
Possible problems with democracy that come to my mind are from the Public Choice literature and other economic models of voters and politicians. Suppose, for example, that people have a good sense of what they want and politicians claim to represent these preferences. But what if people cannot fully monitor what politicians do (and not do)? Politics is pretty complicated, and only the politicians themselves have all the information that allows to assess whether they have really represented their constituency or some other (special) interest.
Politicians could also be tied too closely to their constituency and follow only their interests, thereby undermining the provision of public goods for a broader majority. Or politicians do generally valuable things (or policies with short-term benefits and long-term costs) only shortly before elections.
Some Empirical Flesh
In recent years, social scientists have started to tackle questions on causality, like: Which forms of representations work best, by doing experiments or looking for forms of natural experiments. There is already a lot going on, and I just randomly pick some ideas and insights that might be beneficial for the discussion.
The question whether autocracies or democracies work better for the people is probably impossible to answer in an empirical fashion, because democratic and autocratic states differ on a lot of dimensions.
Hainmueller and Hangartner  look at naturalization decisions in Switzerland and compare municipilaties where all the people decide to those where an elected council decides on who may immigrate. They claim that direct democracy clearly causes fewer positive immigration decisions. They critize this fact as direct democracy then might lead to discriminatory behavior. But what they basically show is that direct democracy works better than representative democracy in representing people's preferences.
Fowler  looks into the adoption of compulsory voting in some Australian states. He demonstrates, as is generally found, that with voluntary voting only, wealthy people are more likely to vote than poorer people, which could distort the representation of interests by electoral institutions. However, he states that compulsory voting causes poorer people to vote and also triggers a change in policy, namely pension spendings.
I think that these pieces are generally encouraging. They have a limited scope, but support the notion that differences in electoral institutions matter.
However, things can also go wrong (but possibly in a good way). Meyersson  claims that electing islamic mayors in Turkish villages lead to broader education of women. That is a good thing, but probably was not what (conservative, islamic) voters wanted from the mayors they elected. He gives a nice explanation for this counterintuitive fact. For our purposes we just note that democratic institutions might have huge impacts in unintended ways.
 Wittman, Donald. "Why democracies produce efficient results." The Journal of Political Economy (1989): 1395-1424.
 Hainmueller, Jens and Hangartner, Dominik (2013), Does Direct Democracy Hurt Immigrant Minorities? Evidence from Naturalization Decisions in Switzerland (January 15, 2013). MIT Political Science Department Research Paper No. 2013-1. Available at SSRN: http://ssrn.com/abstract=2022064 or http://dx.doi.org/10.2139/ssrn.2022064
 Fowler, Anthony (2013). "Electoral and policy consequences of voter turnout: Evidence from compulsory voting in australia." Quarterly Journal of Political Science 8.2 (2013): 159-182. Available at http://projects.iq.harvard.edu/files/westminster_model_democracy/files/fowler_compulsoryvoting.pdf
 Meyersson, Erik (2014): Islamic Rule and the Empowerment of the Poor and Pious” , Econometrica, Vol. 82, No. 1 (January, 2014), 229–269. Available at http://erikmeyersson.files.wordpress.com/2014/02/meyersson_islamicrule_ecta.pdf