It shifts the problem elsewhere, though as you suggest it's sometimes still the same people having to deal with it! For example, anyone relying on celestial navigation who had until now been satisfied with up to 1 second of time error is going to need to start keeping track of UT1-UTC before it starts exceeding those old bounds. This will be possibly within a few years, whenever the next leap second would have been inserted/removed.

A chronometer off by one second gives an approximate error of 0.25 nautical miles at the equator. Meanwhile the general precision of celestial navigation is 1-5 nautical miles. If you need high level accurate positioning, celestial navigation is already not an option.

In the open seas, celestial navigation is more than accurate enough. Once you are closer to shore, where the greater precision is needed, better options exists. Lights, ranges, radar, visual, etc.