Given you have an ordinal response using the function MASS::polr should be more appropriate; it implements a proportional odds logistic regression routine. A very comprehensive tutorial on the analysis of ordinal response variables can be found here. It is also worth checking this thread on: How to understand output from R's polr function (ordered logistic regression)?. In brief, a proportional odds model instead of modelling the probability of response in a particular
category, it models the cumulative probability that the response is not greater than a chosen category.
Your understanding that a binomial family would be too restrictive is fine. A Gaussian with an identity link would be quite unnatural though too; you could not easily constrain it to positive and/or integers responses. You might want to consider using a GLM with a Poisson family but that it is a bit hand-wavy as you need to often define arbitrary categories (so something like glm( ..., family = poisson)). I have seen this being used as illustrative example in some case (eg. Faraway's Extending the Linear Model with R, Chapt. 4.5) but I think it is a bit suboptimal when compared to a real proportional odds logistic regression.
A free and accessible paper on the matter is: Regression models for ordinal responses: a review of methods and applications. from Ananth & Kleinbaum; it is a bit dated (1997) but it is nicely presented and will bring you up-to-speed it relevant terminology.
