I would have been glad to have lived under my woodside, and to have
kept a flock of sheep, rather than to have undertaken this government.
As a native speaker of AmE, I understand would have been glad to refer to a time in the past, not to a present state. But I'm not sure a speaker of BrE would understand the full sentence as I do, or even if a southern speaker of AmE would see it as I do. Perhaps @StoneyB could offer an answer?
I have a little trouble with to have lived and to have kept and to have undertaken in the Cromwell quote.
I would have been willing to drive you to the train station, but you
wanted to walk.
The reference there is to a time in the past when the speaker was willing to drive someone to the train station, but that person wished to walk.
The perfect infinitve seems unnecessary or inappropriate if Cromwell is describing a prospect that seemed appealing to him.
I would have been glad to live... a past prospect
With to have lived I understand Cromwell to be saying that there was a time in the past when the thought of being someone who had lived a bucolic life seemed pleasant to him, and preferable to thought of being someone who had undertaken government. So, there was a time in the past when he was envisioning himself an old man looking back upon his life.
That is, it is not the prospect of living a bucolic life but the thought that he would have been one who had lived a bucolic life that seemed agreeable to him.
Cromwell seems to be describing a past prospective retrospective.
The perfect infinitive might not have the temporal meaning I'm attributing to it here, and it is simply a case of congruence.