Used-To-Be Genji main here. Currently Mid-To-High Diamond in all ranks, if that matters. I’ll try to give a less biased review of the Genji problem.
First of all, does he need to be looked at by the balance team?
In the current meta? No. At most some of his more infuriating interactions should get toned down while adding QoL here and there.
Why?
Balancing Genji is like walking on a very thin rope, even a tiny buff that brings him over to slightly overtuned will be squeezed for all it’s value by good Genji mains, to the point where increasing his damage Anywhere without heavy nerfs elsewhere would make him a must pick IMO. Maybe not so much in lower ranks, since unless you’re smurfing Genji is a nigh throw pick there, but Top 500s would have a field day with even buffing his Shurikens damage by one.
The reason as to why you see so many complaints about him varies depending on the POV: Of course there’s the folks that believe he should get buffed because he’s their OTP and is one of the highest skill ceiling heroes. Let’s ignore those.
Others lament his lack of impact against sustain, sometimes I find myself falling in this group though it’s an opinion driven by frustration more than anything else. Genji used to be able to Solo Dive Supports by taking longer flanks. Thanks to higher, overall easier to land sustain along with the Support Passive this is no longer possible. It doesn’t make him weak at all, just necessitates a mindset shift. Within OW2’s Ecosystem he’s there to pick off isolated targets or focus down whatever the Tank is shooting at or even better, whatever is healing what the Tank is shooting at.
Then there’s the other end of the spectrum. Supports and DPSs without sustain that find him frustrating to kill and get killed by. It is unfortunate, but that’s how the game goes. Frustration should not be cause of balance changes.
To address some of the comment’s complaints about his abilities:
Deflect as an ability quickly looses effectiveness the higher up you go in rank, where mind games are the main focus. Everyone anticipates a Genji to have deflect ready on engage and to proc it once he gets shot at, so the Genji’s job is to catch the enemy offguard with it. Should it receive the changes proposed by OP? No. I’ve seen the argument of Genji being good enough in close range where it isn’t necessary (Which, especially against melee heroes, really isn’t the case), but it’s more of a slight survivability boost than anything worth writing home about.  Good Genjis will waste your time by tanking one or two hits and dash out. It’s up to you to gauge whether the Genji will keep engaging afterwards or GTFO (Hence no need to spend time on them). Also, fun fact, Deflect Does have a cast time. It’s a couple of frames at most, but it’s lead to a lot more deaths than I’d like to admit.
Swift Strike and it’s reset is what makes Genji viable in the first place. Touching any of it’s fundamental mechanics would be akin to what happened to Doomfist with the switch over from OW2 Beta to OW2 Release. Dash in, get a kill, get out is Genji’s gameplay loop. Altering the reset on kill, the ability itself etc would fundamentally make Genji worst to play, both from a fun and viability prospective.
Another complaint many have is Nanoblade, and yeah it is an issue but it’s more a Nanoboost than Dragonblade problem. It’s as problematic as Nano Ram.
Tl;Dr If you want him buffed, git gud. If you want him nerfed, try to change your approach to engaging him. If you’re getting stomped by one, blame Bliz for putting you up a smurf or curse his supports for enabling him. He can’t do jack without team support.