The Routburn might not climb above 1200 but surrounding peaks are 3000 plus. By mid May they might have snow on them but not always. The biggest risk is a small dump followed by a slight thaw then freeze then a second dump. It can happen that early in the season although not often. The date is set arbitrarily by Doc though and isnt changed year on year. Some years there has been no snow well into June others there has been a significant avalanche hazard in March (although by April it was gone)
Closed can have several meanings. In this case closed means Doc stop taking bookings and put the track into a safe condition for winter. They take out the bridges that are on flood and avalanche paths so they dont have to get new bridges each summer and dont have to go hunting for the old bridge. There is nothing to stop you tramping this track at any time of the year but you are going into a place that has hazards not normally found there in that some of those streams are uncrossable in flood and because of the conditions can change very quickly from good to killers. If you get caught between 2 flooded streams and the hut isnt in that section you have to be able to shelter for up to several days so a tent plus extra food becomes a must have
On great walks expect any river that might possibly get water in your boots during summer to be bridged. The time you are looking at tramping if the weather is good you might get your knees wet. But also consider that if the weather isnt good they could become quite difficult or worse.
Ive crossed rivers up to places that shouldnt get wet and swum a few crossings but the deeper the water the slower it has to be going to be safe. Ive turned away from crossing not much more than my knees.
So yes an unbridged stream could easily be unpassable especially traveling solo.
Its a risk you take or not. The 2 biggest killers in NZ bush are river crossings and weather
If the day you cross the Harris Saddle happens to be a clear day, please allow for - and do! - the side trip up to the Conical Hill Lookout. In my humble opinion, the greatest view (of so many!) from anywhere on the Track.