I'm sorry that article is absolute tosh. Please don't use a tabloid paper as a source of professional information
For starters, there would be no way of knowing if you had an oily nail BED or not lol. The nail bed is inside your finger, under the nail.
The nail PLATE, which is what we are concerned with, is rarely 'oily'. It is not common to have nail plates with an excess of oil, but for most people there will be natural oils present from things that you come into contact with, and a small amount that is naturally produced.
This does not cause nails to lift. Improper prep of the nails causes enhancements to lift, such as poor cuticle removal and improper dehydrating and cleansing.
This is why proper training, and correct use of products is important, as well as sound professional knowledge, such as using the correct terminology.