Interpreting correlation between fixed effect and explanatory variable
You have done two different things.

Your fixed-effects model captures the within-group over-time functional relationship between $debt_{it}$ and $y_{it}$ (that is, how much average difference in $y_{it}$ is there between two periods with a 1-unit difference in $debt_{it}$ within a country). In your data, there is limited within-group variability in $debt_{it}$, which probably lead to a large standard error and a resulting insignificance.

Your reasoning about the correlation between the fixed effects and the explanatory variable is about the cross-sectional functional relationship. You found that countries with higher $debt_{it}$ have higher $y_{it}$ values (as opposed to periods with higher $debt_{it}$ having higher $y_{it}$ within a country). This is not what you originally intended to investigate when you set up your fixed-effects model.

Your Hausman test says that the within-group functional relationship and the cross-sectional functional relationship are different.

That said, yes, it is true that "a higher debt does result in higher interest rate spreads", but it is true only in the specific sense that "countries with higher debts show higher interest rate spreads". You should be careful when saying "result in".

Overcoming the issue of limited over-time variation and having significant results is hard. There is no magic. Abandon the fixed effects model, and try to control for many time-varying and time-invariant regressors, enough for you to argue that you controlled for most country-specific factors. (You can use RE or POLS estimation.) People might still criticize that you didn't control for enough factors, but you will need to defend yourself somehow. You could also move over to dynamic models or others, but that's a different story.