だが、概ね天皇は世俗権力を追求することなく存続してきたようです。

、。

It does appear, though, that the emperors have on the whole persisted without chasing secular power.

Literal

But, on-the-whole emperor [topic-は] secular-power [obj-を] pursue [thing-なく], has-continued-existing [seems-ようです].

概ね 'generally / on the whole' is a formal written adverb for broad generalizations. 世俗権力 'secular power' is a sinitic compound contrasting with 宗教的権力 'religious power.' 追求する 'to pursue / seek after.' ~ことなく is a literary negative conjunctive 'without X-ing,' formed from the こと nominalizer + the literary なく. 存続する 'to continue existing / survive / persist' is formal vocabulary. ようです softens the claim as an observation.