What was the phrase clean-slate originally used for? Or did it always refer to restart in something?
Clean slate comes from the time people wrote on (slate) chalk boards. Whether that was your tavern bill or your letters in school. Starting with a 'clean slate' meant you wiped the board clean and started over.
Clean slate comes from the slate boards used in schools. The expression is used figuratively from the mid. 19th century.
A fresh start; another chance after wiping out old offenses or debts. This idiom often appears as wipe the slate clean. For example, Henry's boss assured him that the matter was finished and he could start with a clean slate, or He wished he could wipe the slate clean, but it was too late to salvage the relationship.
This expression alludes to the slate boards on which school work or tavern bills were recorded in easily wiped-off chalk. Since 1850 or so the term has been used figuratively, and it has long outlived the practice of writing on slate.
The term "clean slate" was originally a term which referred to clearing debts. Whether they were social debts such as poor grades in school or financial debts such as at a grocery store, the "public ledger" was often slate marked with chalk. A clean slate relationship, therefore, meant that past debts were erased and two people started on equal footing.
As chalkboards moved into classrooms the term was co-opted to signal restarting a conceptual pursuit. If an equation, design, or taxonomic problem was too complex to handle one would simply "clean the slate" and start again.
Note that in both original etymologies the term is somewhat dichotomous. Although an optimistic interpretation of the idiom implies starting over with no record of the past, it is realistically quite difficult to completely clean a slate before starting over. When trying to start over on a clean slate, one often is actually writing over smudges of past mistakes.
In modern parlance both meanings live on. In particular the term clean slate has seen a recent uptick due to its use as an engineering term. A "Clean Slate Design" or "Clean Slate Initiative" such as Boeing's 787 and Project Yellowstone, respectively, imply throwing away old engineering assumptions and designs. These terms are quite popular in aerospace and defense realms where in some cases the slate has not been cleaned since the beginning of the cold war.
Some have pointed out that the term is especially appropriate in engineering circles as engineering projects are generally rife with technical debt and engineering organizations often struggle to leave behind old habits or systems. I must apologize, though, as I'm not currently able to find or cite where I read that.