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.
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.