See this thread which discusses that Avraham Avinu used to make his guests pay exorbitant prices if they did not wish to bless Hashem for the food.
I am struggling to differentiate between a missionary from a different religion who would set up a food kitchen in order to persuade people to convert to their religion, and Avraham's hospitality.
To me, it appears that if the goal of the generosity is to convert people to a religion, then the deed is not Gemilas Chassodim, but rather some sort of avoidah, an act between a person and his God.
In fact, one can go as far as to say the food was not free at all, but rather had a payment of blessing a god you do not wish to bless.
Wouldn't someone who is a ba'al chessed give food to anyone who is hungry, and not make it dependent on recognizing their God?
(PS. On a similar vein, Avrohom sent Yishmoel away without gold and silver according to Rashi, as he hated him because he went on a bad way. Wouldn't someone who is a ba'al chessed be generous to someone who had stayed in his home for years, regardless of his religious activities?)
In summary, I am seeking to understand where Chazal saw that Avrohom Avinu did chessed from these accounts, or are there other stories that were purely chessed from which Chazal learned from?