... you get the strong impression that the 'thorn in the flesh' must have been something to do with infirmities, i.e. physical issues of sickness or weakness.
That's very much my take on it too. I have wondered if it was in some way shameful to Paul and perhaps connected to his own actions in youth or an inherited more substantial defect than something such as a weakness of the eye etc (though this could be one of the symptoms of a wider more systemic issue). The term 'thorn in the flesh' is amplified by it being 'the messenger of Satan'. The use of the word 'flesh' so close to the naming of 'Satan' may have connotations of having at one time in his past succumbed to the hidden drives of the flesh i.e. lust or even perhaps being the offspring of such a union that led to issues post birth. Syphilis, either inherited from the mother or contracted by the individual can lead - when untreated - to significant long term problems that riddle the body with perpetual pain and discomfort. For example on inherited Syphilis:
Signs & SymptomsCongenital syphilis is acquired by the fetus when the treponema pallidum spirochete is present in the mother. Pregnant women with syphilis may have a reduction in estrogen while serum progesterone levels may increase. Symptoms of early congenital syphilis usually appear at three to fourteen weeks of age but may appear as late as age five years. Symptoms may include inflammation and hardening of the umbilical chord, rash, fever, low birth weight, high levels of cholesterol at birth, aseptic meningitis, anemia, monocytosis (an increase in the number of monocytes in the circulating blood), enlarged liver and spleen, jaundice (yellowish color of the skin), shedding of skin affecting the palms and soles, convulsions, mental retardation, periostitis (inflammation around the bones causing tender limbs and joints), rhinitis with an infectious nasal discharge, hair loss, inflammation of the eye’s iris and pneumonia.
Symptoms of late congenital syphilis usually present themselves after age five and may remain undiagnosed well into adulthood. The characteristics of late congenital syphilis may be bone pain, retinitis pigmentosa (a serious eye disease), Hutchinson’s triad which is characterized by pegshaped upper central incisors (teeth), and interstitial keratitis which consists of blurred vision, abnormal tearing, eye pain and abnormal sensitivity to light, saddle nose, bony prominence of the forehead, high arched palate, short upper jawbone, nerve deafness and fissuring around the mouth and anus.
The term 'thorn in the flesh' could be code for sexuality as well as constant pain and irritation that never leaves one.
Syphilis has also been seen by mystics and sacred poets as a metaphor for the cost to the spirit of taking part in material existence with all its devilish traps. Shake-speare for example famously concluded his sonnet sequence with a double set of highly enigmatic verses that relate to the wisdom of finally unifying an understanding of fire and water, and the 'gifts' of the gods Apollo and Artemis, with having contracted the ever burning fire of Syphilis that can only be balanced by the gift of quenching spiritual water.
Pure conjecture and probably way off beam.