I find the stems far too fibrous to eat and if any leek has even an embryonic stem inside I will remove it before cooking.
It would be OK for flavouring stock or something as long as it was strained out before serving.

You know you don't need to get rid of them.

For one thing, each of those seed heads will provide enough seeds for probably a few hundred baby leeks. They often self-sow with wild abandon, look for little lawn-like patches of baby leeks to emerge in the spring.

They also regrow from the original bulb. In another roughly 6-8 weeks, one to four or five little baby leeks will emerge from the base of the old leek. They will be big enough to use yet this fall as small leeks, and will become full sized next spring. Once the flowers fade and start to develop into a seed head, you can pull or dig the entire plant if you want to. You will see the small cloves without a tunic (papery skin) clinging to the old stalk, attached at the basal plate. They look like peeled garlic cloves. Each of those will grow into a new leek. You can dig these, separate, and replant individually if you want them to get bigger. Left on their own to clump they will still be OK, but as large as if separated.

Finally, you can also use that little clove as is as a garlic/onion substitute. Just wash, cut off the basal plate, and slice, dice, put through a garlic press, whatever.

As Flora_UK wisely said, you can always use the old stalks as a "bouquet garni" to flavor soups and other dishes, just remove before eating (unless you're a porcupine or beaver that enjoys woody stalks).

So, yes, there is plenty of life and usefullness in a "spent" leek.

denninmi - that is really interesting. I've read your posts before about self sowing and self increasing leeks. The strange thing is that this has never happened in my garden. I have one old leek which I have been watching for some years. It produces a flower, which I leave, but it never self sows. It produces a single off shoot which perpetuates it but never more than one. I wonder why that is?

I left mine to flower this year cause they really didn't get very big, but then got woody.
I have left some to go to seed, but never saw any pop up the next year.
I'm letting carrots, leeks and onions do their thing this year. We shall see! Nancy