Ask Mr. Smarty Plants

No, cutting off the stalk of buds before the agave blooms is not likely to save the plant. The reason Century Plants are called that is that it takes them up to 40 years (but not a century) to bloom. And the reason for that is that the agave is native to very dry and forbidding desert areas where water is scarce, the sun is unforgiving and the soil not much better. Every plant is driven by its own genetics to reproduce those genes. In order to do that, every plant must bloom and manufacture seed. This takes an enormous amount of energy; by the time you see the bloom stalk emerging, the plant is already on its way to dying. The blooms represent many years of work to reproduce, and if you cut off the bloom, you lose both the incredible sight of a blooming Century Plant, but also the plant, which you were going to do anyway.

In our Native Plant Database, there are nine plants with the common name "Century plant." All are members of the genus Agave, and not a single one is native to Florida, or even close. Agave americana (American century plant) has these propagation instructions on our website page:

Propagation

Propagation Material: Seeds
Description: Division by offshoot of pups, seed
Commercially Avail: yes
Maintenance: Removal of old lower leaves or dead plants can be difficult due to size and leaf tip spines.

So, you see, you answered part of your own question when you mentioned the "pups" around the present plant. You can take them out and transplant them now or wait until the blooming is ended, so you don't have to worry about damaging the blooming plant as you dig out the offspring. Just heed the warning to be careful about where you transplant those babies, they will not always be babies. When they grow up, they need to be somewhere that they will not hurt passersby like your children, your pets and yourself. As they get older, they are very difficult and dangerous to move. 

From the Master Gardeners of the University of Arizona, Pima County Cooperative Extension, we found this information page on Century Plant. From Floridata, of all places, more information on Agave americana.

To quote from one of our own previous answers:

"Agaves produce new smaller plants around their base. All you need do is remove the pups from the mother plant using a trowel or knife and put them in smaller pots with the same kind of soil mixture that your original plant has been thriving in.  If you don't know what the original is growing in, nurseries carry "cactus mix" potting soil which is grittier and more like the desert ground the plants are used to. Keep them watered, but let the soil dry a bit between waterings so they don't rot.  These pups can have very long roots that connect them to the mother plant, but you can break them off to about the same length as the height of the plant or whatever will fit in your new pot.  Even if you think you have lost too much of the root, pot it up anyway and see what happens.  Agaves are very hardy and forgiving plants!"