An important point here is that the integral of the likelihood function over different parameter values is not constrained to be 1. This is why a likelihood is not a probability or a probability density, but its own thing. The confusing bit is that the likelihood formula is exactly the same as the formula of the original probability density function...
Splitting hairs probably but personally I'd say it the other way around: a likelihood is not a probability or a probability density, so there's no reason to think that it would integrate to 1.
The reason it's not a probability or probability density is that it's not defined to be one (in fact its definition involves a potentially different probability density for each point in parameter space).
But I think I know what you're saying -- people need to understand that it's not a probability density in order to avoid making naive probabilistic statements about parameter estimates or confidence regions when their calculations haven't used a prior over the parameters.
