It is obvious that ELISA is time consuming. It will be the preferable method when you are considering cost.
PCR got advantage in the sense that it can be design to target more than one virus in one set-up and is sensitive to catch virus with low accumulation or few virion in host tissue. Bear in mind that nucleic acid extraction is equally a task to overcome. For the purpose of a sound research design both should be use OR one of them complemented with different detection method...bio assay, microscopy....
ELISA is more sensitive than PCR because immune tests using mAb with no cross reactivity gives accurate detection comparing with PCR that more likely to get a contamination.
PCR is more reliable than ELISA. This is because, when I used ELISA, there were some false positives while detecting potato viruses. Further, sometimes ELISA results are not conclusive enough since they can show no infection when the viral load is low. This is a serious phytosanitary threat especially in seed potato tuber trade where a consignment may be declared disease free. However, after time, the pathogen may accumulate to levels high enough to be detected by DAS/NCM ELISA which can lead to pathogen spread. I would recommend PCR.