It's all about minimums.
To be ILS or LOC it has to have different minimums for a "localizer only" approach, it is not all that common to have an ILS without other minimums but we have one here at MSO. If you lose any part of the ILS 11 approach you can't do it. You are correct in that the localizer signal will be there and you could navigate with it, but, how low could you go? where is the MAP. The key is how many different minumums does the approach have for different equipment having failed. The LOC mins will be higher than the full ILS mins.