The seeds will definitely grow, and being from grocery store, or other source has little affect if any on germination.
My grandparents who were born in the 1800's were farmers and did have an orchard.
Here is what My mom told me many years ago, since she had a huge apple tree in her yard. (probably equivalent to the one you had Chickendreams)
Planting an apple seed will not produce a tree bearing the same apple as seed came from. It will produce something with whatever genetics are in it. (MIXED). She said, (and this was old school estimates) that one in one hundred will be a genuinely useful apple tree. The rest will be wild trees that will produce usually small apples, with varying flavors. With this being said,,,, my grandparent would plant apple seeds , and then graft them with twigs from their choice apple trees.
The same thing is done with rose bushes. (not all, but choice ones)
The root stock is grown from hardy wild varieties. They are then grafted with the choice variety. When you purchase rose plants , it is easy to see the bud union. When your rose grows, you only allow new growth from above the bud union. If you allow twigs to sprout and grow from below, or from roots of that rose,,,,,, The rose will go wild, and most likely not produce desired flowers.
I had some roses go wild in my flower beds. They grew like 7734, but did not produce any flowers

.
I can most certainly understand the sentimental value of your apple tree Chickendreams. You can still maintain that by planting a seed or grafting a twig from it. Those are the feelings that will stay in you Heart .

forever. and renewed.
About 30 years ago, mom and dad planted a few plum tree in my yard. They were from plums that they kept in their yard. Only one original plum tree is still left from that planting, but so many new trees sprouted. I have them all over. I also shared the new sprouts with with my children, to grow in their yards. The memory just goes on with it.

. I shared it with neighbors as well, and the story.
Here is a pix of the plum tree in spring bloom.
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