It's not about completing the task, it's about addressing it on the day it appears and doing what is necessary related to that task just for that day so you know you are done with that task for 'today'.
Then when the new day starts, all the strikethroughs are gone and you start over.
In other words I'm not completing the task. I'm just striking it through on the 'today' tile to indicate I am done with it for today. If it appears again tomorrow fine, but I would like to be able to go down my checklist of tasks for the day and strike them through until there are none left.
Just a thought... A partially transparent strikethrough line might work best if that's possible (without rewriting TDL from scratch that is!

Here is a diagram of what I mean.