Hi RESORT2ME, for the best and largest flowers, IF you were going to attend a show with the plants, to get the best from each ( flowering stem), you would remove all but the best looking flower bud BEFORE they got the chance to grow larger than say a marble or your small finger tip. this is because all plants use a lot of energy by production of flowers therefore show flower growers want all the plants energy go into prize winning flowers.
As we are all amature gardeners, the only time I would remove any flower BUDS is IF the plant had say 2/3 buds all growing like a bunch or maybe turning brown and spoiling the show BUT, Peoney's have such large flowers and by nature / design or whatever, lot's of plants only produce enough flowers they can cope with as they don't want to be weak and die back before the flowers are done, seldom do you have to remove buds in the making, however, IF the flowers begin to fade or die, then it would be best to remove these dying flowers IF there are more buds still to open out into flowers and this will extend the flowering season.
Most annual plants will produce many, many flowers heads as there sole purpose in life is to germinate, grow greenery, flower, and send out all there seeds in the same year, Perennials and peony's are in that group are not in the same rush to make as many flowers as these plants make new flowering stems from there tuber under ground.
Hope this helps you understand a little better and you enjoy those beautiful plants and their massive flowers.#Best Regards, WeeNel.
The benefit for the plant is possible reproduction by forming seeds... But the benefit to you of leaving the spent flowers alone to form seed heads is that the seed pods are rather attractive and interesting... plus the chance to get more plants from seed. This depends somewhat on the sort of peony you're growing - some cultivars with very high petal counts have been rendered unable to form seed. If you're growing species or single or semi-double flowered cultivars, normal seed pods can form - when they split open, the black fertile seeds and red infertile seeds are quite attractive.
Errr... When buds have already formed, they will bloom whether or not you cut off the other flowers...
You are a lucky gardener to have a peony that gave you such a wonderful show, to be honest, don't go by that standard for every year, I have tremendous years for great blooms from lots of different plants and maybe the following year, not quite so good, I think temps, soil conditions, rainfall etc all play a good part long before the new shoots appear early spring.
As regards cutting the whole stem off the plants once the flowers have died away, you can either cut just under the flower head and leave all the remaining vegetation /leaves/ stems to die down naturally as bulbs and tubers do take goodness from this slow decaying of the foliage, or cut away the full stem but leave about 3 inches above the tuber for safety.
All perennial plants are best left to decay slowly with only flower heads removed but, as you do the dead-heading you need to check all the plant over and remove any diseased foliage as you don't want that type of problems going into tubers or bulbs or spreading further along your border / bed, only to return and weaken the plants the following year. Also remember IF you dont do a tidy up when the dead stuff is laying on the soil, it can harbour slugs/ snails and other nasty things that are detrimental to our plants.
At the stage of Dead-heading by the way, that is when I add a handful of either chicken manure pellets or a multi purpose dried feed and gently rake this in around the root area and with Peony's this could be twice as wide as when you planted it and it increases each year normally, this helps the tuber take up this goodness to strengthen next years plant growth.
either way you will get more confident and not worry too much about your gardening skills, as we all did, we learn by practice, advice from others and general common sense in lots of tasks. Well done to you and hope you have enjoyed watching this fat ugly looking tuber turn it'self into such a thing of beauty, nature does things way better than we can eh.
Best regards, WeeNel.
Another thing I would like to add to what WeeNel said, is that peonies can harbor Botrytis blight, which can be carried from one year into the next. Because of this, you should cut back your peonies in the fall after all the foliage has turned brown. The stems and leaves should be removed and discarded. I know one year that was particularly rainy and damp, I did have a problem with Botrytis blight. That year I sprayed with an antifungal spray and removed all foliage in the fall as mentioned. I have not had a problem since.
I typically wait for spring cleaning to get into my garden clean up so the birds can enjoy the seeds from my flowers, however - the peonies are cut back every November.