As far as I can tell, if reeds are burned in your world, they do not come back. The wiki and every other source I've seen doesn't say anything about reeds being in the regrowth mechanic. I have read of people who had their reed farms burned during summer, feeling they lost the reeds forever, and that really sucks. I bet this will happen a lot more with Wurt's release because swamp bases will be more popular.
Whether reeds are renewable through world regrowth mechanic, or reeds produce seeds you can plant, I think it's important they don't go extinct. Every summer that you are not in the caves or the oasis biome, things around your base that are not covered by an ice flingomatic will smolder and burn, and the fire spreads rapidly. If you decide to base in the swamp (which I assume a lot of people will be doing with the release of Wurt), then the reeds in your world could go extinct by wildfires and various items smoldering. Reeds do not burn off screen, but if you base at swamp they will be on screen every summer, and because of that the reeds nearby will likely eventually burn to ash and go extinct.
Another separate suggestion, and not as important, is to make reeds diggable and plantable on swamp and pebble beach turf. This would make the reed trap set piece not so desirable and luck based on getting that world generation, because you'd be able to make your own reed farm. One reason Wurt looks cool is because she can make her own swamp by crafting swamp turf and crafting merm houses, but the swamp that she makes will never have reeds because there's no way to remove them or for them to regenerate on swamp turf.