Here are a few points to consider...
It's generally a rule of thumb that the ability to destroy will always be easier than the ability to defend. Thus arms races exist, typically with defensive measures losing to increasingly capable offensive measures.
Guns are advantageous because they offer ranged capability to destroy.
Defensive capabilities in the Mass Effect universe (and other sci-fi/fantasy settings) are capable of mitigating that previous rule of thumb I mentioned. They are at a technological point in their arms race where barriers, shields, etc, can completely negate certain types of damage.
Vanguards have this thing called Biotic Charge.
Your primary advantage of having a gun, range, is entirely useless against an enemy that can instantaneously close the distance. Or has strong enough shields/barriers/regenerative health to sustain your incoming fire and still be capable when he/she/it has closed to c.q.c.
Now think about that sword that's being swung at you instead of a bullet fired from across the map.
If that sword that's being swung at you happens to have biotic enhancements like say the Krolord's hammer in MP what do you think is going to do more damage?
A sword swung by a cybernetic and biotically enhanced ragemonkey of a Vanguard, or single shot from let's say a mattock.
Really grasp this, all the benefits of a gun are potentially mitigated by things that already exist in the ME:U.
For melee to be as unfeasible as some of the anti-melee posts I've read, guns need to get a serious buff, and people need a way to counteract beings that can Biotically Charge themselves across a field.
Show me any blade, swung by a person, cutting through modern combat armor.
Arguments like this, you aren't dealing with today's technology, you're in the Mass Effect Forums talking about a couple hundred years in the future.
You aren't talking about a person, you're talking an enhanced human with biotics, cybernetics, and god knows what else.
You aren't talking about a "Blade" you're talking about a 1.5m long singularity, or an omni-molecular blade that could cut through armour without any pressure at all.
***
The problem with melee isn't it's feasibility, it's already here, it's already done. They just need to expand on it, and balance it. And balancing it will be the hard part.