1 sat/byte might result in your transaction getting stuck during high demand times. so you should rely on the slider instead. the further to the left the lower the fee and the slower the confirmation times.
There are some perfect time periods to move coins each day with low transaction fees. At fees at 1 satoshi per byte, you can move it well if mempool is empty (like o_e_l_e_o mentioned). If you do your transactions when fees stay at highs, and you set your fees at 1 satoshi per byte, you will have to wait hours or days to get confirmations, that depends on mempool's emptiness. Your transactions in those cases will be stayed in queued (with priorities for higher-fee transtions, descending to 1-sat-fee transactions, include yours). If you don't want to have nearly instant transaction, just set fee at 1 satoshi and wait.
you should note that "empty" in this context is not a clear term to use since mempool of bitcoin can never be actually empty. it is a term used when people mean "mempool has lower number of transactions". but then one might ask how many is considered "empty"? 1? 1000? 2000?
the answer is in the sum of transaction sizes (weights to be accurate). if that sum is smaller or equal to block weight then you can say paying the minimum is high priority.