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Spring-blooming trees and shrubs are usually pruned right after they bloom - no later than June 30. It is acceptable to prune when dormant in late winter but, of course, you'll be removing some blooms. The advantage of pruning a tree before it leafs out is being able to see the overall structure and the branches needing removal. The choice is yours. Dead or diseased branches should always be removed as soon as possible.

The Bradford and other varieties that give off bad smells are part of the Callery Pear grouping, Pyrus calleryana. There are many other species of pear that do not give off this smell, including Pyrus communis, the species that includes the Anjou and Bosc varieties, and Asian pears. I am not sure whether there are specifically ornamental pears that don't smell bad, but the edible pear varieties are also beautiful trees. Here is more about pear varieties:

https://www.gardeningknowhow.com/edible/fruits/pear/types-of-pear-trees.htm
https://www.gardeningknowhow.com/ornamental/trees/ornamental-pear/ornamental-pear-trees.htm

There could be a disease in the soil, or environmental issues that have had an impact on the tree. Here is a link that can help: https://www.gardeningknowhow.com/ornamental/trees/ornamental-pear/bradford-pear-not-flowering.htm

Bradford pears are within the group of flowering pears. There are many flowering pears. Callery Pear is another that looks very similar, but is less prone to breaking.

The most common pest on a Pears, is the Coddling Moth.
This article has more information.
https://www.gardeningknowhow.com/edible/fruits/pear/fixing-pear-tree-problems.htm

This looks like a fatal injury to me but only a certified arborist, who can inspect the entire tree and how deep the injury goes, is qualified to answer your question. Repairing the damage here and bracing other at-risk limbs may equal cost of removal. Bradford pears only live 18-25 years before they decline due to old age. Call in a pro if you want to save your tree. Consider replacing, if necessary, with a hybrid pear, Pyrus calleryana, ‘Aristocrat’. Concerning pruning: there are select times of year when pruning is optional or routine. But when there are dead, damaged, diseased or dying branches they should always be removed ASAP as they leave the plant susceptible to infestations/infections.