Leaving the question aside as to to what extent the alleged pitch element of the Latin stress was borrowed from the Greek, I want to touch on the exact nature of the stress, in terms of pitch contour, something which I have pondered for a while now.
"The circumflex accent is thought to have has a slight up-down tone, the acute a straightforward upwards tone. Final unaccented syllables had a slight falling tone. (This is called the grave accent, but this is not written, it is simply understood to be there.)"
I wonder if there's not another interpretation of the evidence, which would amount to a somewhat different description:
A lot of the ancient testimonies on the Latin pitch accent are collected in The Roman Pronunciation of Latin
by Frances E. Lord. I'm not going to quote them all here, but the terms which are used to describe the acute accent are "acutus; acuit; acutus tenor; elevat", terms pertaining to the grave accent are "gravis; deprimat; deponat", and the circumflex is described in terms of "(circum)flexus; flectit", and in general as being a combination of the acute and the grave. Lord summaries it in English as "In Latin the acute accent means that on the syllable thus accented you raise the pitch; the grave indicates merely the lower tone; the circumflex, that the voice if first raised, then depressed, on the same syllable."
Now, neither the English, nor the Latin, is very precise here, I believe. What I am aiming at is that phrases such as "the pitch is raised on a syllable" or "syllaba acuitur" can be interpreted either in the way Evan takes it, that the syllable is started at a low pitch, which then gradually rises towards the end of the syllable, or, in the way I'm inclined to understand it, namely that the syllable is pronounced in a uniform, stable high tone, i.e. that the pitch is raised (="high") on that syllable in comparison with the other syllables.
As for the grave, note that this accent is said to apply to all syllables, not only to the one following an acute, and it can in any case not be that all unaccented syllables are to be pronounced with a falling
tone, but rather with a low
The circumflex accent is a combination of the two, and thus, a syllable pronounced with the circumflex accent is, I argue, supposed to start on a high ("raised") tone, which gradually falls to a low tone before the syllable is finished. I believe it is misguided to consciously attempt to start an acute or a circumflexed syllable with a low tone.