This sounds to me like something best suited to a Poisson or negative binomial model. You have count data - that right there should tell you that linear regression is not appropriate, and because each observation isn't a functioning vs failing widgets, but number of failed widgets per firm. There are panel Poisson models that you can use.
The difference between Poisson and negative binomial is basically whether you have overdispersion or not, i.e. the mean=variance of the counts. There are numerous ways to test for this, that a simple Web search will show you.
The question you ask about proportion vs raw counts is one of offset or exposure. A count model without an exposure variable is interpreted as the predicted number of failures at any given firm, based on the covariates. When you have an exposure variable, say total widgets produced, you enter that as a constrained independent variable, with coefficient of 1, just as you seem to understand in your description of how you would do it in a negative binomial model. The difference in arithmetic is:
$log(failed widgets)=\beta_0+\beta_1IV +\beta_2year$
$log(failed widgets)=1*widgets+\beta_0+\beta_1IV +\beta_2year$
The interpretation is no longer the predicted total number of failures in the model with the exposure variable, but the rate of failures.
Which do you choose? Here are the criteria I usually use:
- What makes most intuitive sense to myself and other stakeholders?
- Does the number of failures make any sense without considering the number of widgets produced in total? Sometimes it is useless to compare failure between large and small factories without considering their size.
- Does a rate make sense? Sometimes a rate is not what you want. If it is for compliance purposes, and producing more than say 10 widgets is unsafe for the cosmic consciousness, then it doesn't matter if its a large or small firm producing. In fact, maybe small firms are more likely to have no failures because they do everything by hand and are more meticulous.
Here is another link that may provide more insight.