Thats a tough question.
Personally if I aquired the cats somehow I would keep them, especially if they were "in memory" of a loved one. But as far as other people I can not speak for them.

My answer would vary due the cats personalities. If I could not keep them forever I would at least try to house them temp. until I could find a home or shelter. Some shelters (especially this time of year) are overcowded with cats so they may not have any room for any more. Which is pretty much the same as a no-kill shelter (I call most no-kill shelters, limited room shelters) as once they reach their limit they do not accept any more pets.
Some cats can adjust to new surroundings easily, some take a little time & some just don't. If it did not work out keeping them at my home or confined & no shelters had any openings than yes euthinasia is better than fending for yourself on the streets.
But finding a vet to do that to a healthy animal MAY be difficult. I have worked at a couple vets office & have brought my pets to other vets offices from NY to Oregon & then in between & up & down & yes there are some that will euth. any pet but you would be surprised at how many do not (which is the way it should be). My current vet will only euth. animals that are patients, no one off the streets can just come in & have their animal put down. The animal must be very ill, injured or be dangerous to others to be put down. Even of your animal has a slight injury or illness our vet will consult w/ you about other possibilites (ex. foster care, money issues, training & behavioral issues, adoption etc...). It has to be the animals best interest to be euth. at my current vets office.