Ice plant (specifically Carpobrotus edulis here) is such a ubiquitous invasive plant on the mainland that eradication seems almost futile. The unique factor in restoring island habitat is that with dedication to eradicating resprouts and taking strong preventative measures for all traffic coming onto the island to be clean of invasive seeds, we can actually completely remove invasives from these delicate island ecosystems.

Why remove ice plant in the first place? I'll let the California Invasive Plant Council answer that:

    Highway iceplant tolerates a range of soil moisture and nutrient conditions and can establish and grow in the presence of competitors and herbivores. These qualities and others have meant that in many natural areas it has formed nearly impenetrable mats that dominate resources, including space. It has invaded foredune, dune scrub, coastal bluff scrub, coastal prairie, and maritime chaparral communities, and competes directly with several threatened or endangered plant species for nutrients, water, light, and space (State Resources Agency 1990). It can suppress the growth of both native seedlings (D'Antonio 1993) and mature native shrubs (D'Antonio and Mahall 1991). In addition, it can lower soil pH in loamy sand (D'Antonio 1990a) and change the root system morphology of at least two native shrub species (D'Antonio and Mahall 1991). - Cal-IPC