Short answer: No, you can't use p-values to "select significant pathways"; you need to use adjusted p-values.
Long answer:
If you're doing hypothesis testing and want to claim "statistical significance", you should use adjusted p-values because of multiple testing. The multiple testing problem persists in exploratory analyses (when you simply throw in a gene list and want to see what comes up as "enriched") because you aren't testing a single pathway, you are testing hundreds of pathways.
Why you're getting adjusted p-values of 1? Perhaps your input gene list is too small so there's no way to get sufficient evidence to support enrichment. Or, perhaps, your input gene list is all over the place (i.e. the genes don't fall into a specific pathway/category) so there really isn't any statistically significant enrichment.
Whether you can extract any meaningful information from the analysis depends on the underlying question (which you should have already formulated a priori). If your question is about: which pathways are statistically significantly enriched (which seems to be your question), then, well, the answer is none. If you had a different question (e.g. are pathways related to metabolism generally more enriched than pathways related to cell cycle), then that's a different story.
Other ideas: to mitigate the multiple comparisons problem, you might want to do some filtering beforehand (e.g. exclude certain pathways or merge similar pathways where nearly all the genes are the same among the pathways). Or, if you have the results of differential gene expression analysis, you might want to want to try GSEA instead to see what pathways tend to be upregulated or downregulated (which works better than enrichr analysis if, say, your differential expression threshold is too stringent). However, these things will require some more work on your end -- you'll have to download the genesets from enrichr's website and do the analysis yourself (not as easy as using a GUI).
