I don’t think you understand why felons are excluded. By being convicted of a felony, a person forfeits some of their rights as citizens. They can’t vote, be elected to public office, serve on juries or own firearms.
It may not surprise you, then, to know I disagree with that in general as well. 
What I don’t understand is guns are mostly readily available in the United States and in most states the requirements to be permitted to carry a gun are not very onerous (in some states open carry requires no permit at all.) Yet in pretty much every jurisdiction in the United States it is absolutely illegal to openly carry or concealed carry a full length sword.
I think if every American carried a saber or rapier or such it could quite possibly decrease a great many crimes. The constant presence of swords would be a reminder that any altercation would very easily escalate to a sword fight which may have disastrous outcome for one or both parties, thus dissuading altercations and leading to fewer assaults and manslaughter incidents.
It takes a LOT of effort to become an accomplished swordsman, and an incompetent swordsman has no chance whatsoever against a competent one.
Compare/contrast it to guns. “God made men, but Sam Colt made them equal.”
You are okay with the idea of convicted felons serve on juries?
I’m not too keen about letting white collar criminals run for office, although I will concede that in Illinois the difference might not be that noticeable.
Great, so we replace general physical education with fencing. 
The Bill of Rights guarantees freedom of assembly; no one claims that this means that police can’t order an unruly crowd to disperse, or that cities can’t require parade permits for demonstrations. The problem is when cities like DC and Chicago pass gun laws that- to continue the analogy- are the equivalent of having crowd control ordinances that pre-define every unauthorized public gathering as a riot.