You're reading way to much into this :)
In order to appreciate this song, I think you need to allow for variation of meaning in the English language. Understanding some songs even require that we grant them some poetic licence, although I think we should treat those with cation because consciously or otherwise those do tend to cloud our theology. However in this case, I don't think poetic license is even necessary to appreciate the meaning.
Debtor could have several meanings. One of them is quite simply the sense of 'thankful', but I think the meaning here could actually be understood in a much stronger sense than thankfulness. The word can carry a sense of being 'dependent' or 'bound to'. And yes, theologically speaking, I think it is fair to say we are dependent on God's grace every day. While we don't own him anything in return for it, the very fact that his grace is freely given does "daily constrain us" to walk in His ways.