"What intrigues me is the first meaning where exemption from sickness is acknowledged"

The problem is you are applying the definition of welfare as applies to the INDIVIDUAL; the constitution is a document about the powers of the STATE:

"Exemption from any unusual evil or calamity; the enjoyment of peace and prosperity, or the ordinary blessings of society and civil government; applies to states"

I find it difficult to believe that the framers of our constitution wanted to grant government the power to interfere as it saw fit for "the greater good". As we know, the road to hell...

As you stated, the founders were very adept with language, and anyone of that generation would undoubtedly know that in the instance of the constitution that the "general welfare" clause meant for the good of the STATE, not the individual.