Lol Any Advice?

As someone that has mained a small group of champs that has been basically forgotten by time since Assassins became the staple of mid you can hit diamond playing anyone if you put in the hours. For instance, Nautilus jungle has been one of my go to picks since I picked him up in season 2 and probably have at least 300 games on him. It's reflected by the fact I have maintained a 60%+ winrate with him in ranked since season 3 and peaked at 70%+ in seasons 4 and 6. I don't think he's been considered a relevant jungler outside of small shimmers of attention but I stick to my boy. Same could be said for champs like Morgana, Sona, and Lux for me.

It's rough though, I know that. Trying to play Morg mid in season 3 and 4 was a chore and I eventually just took her bottom since I'm a support main with jungle as a secondary (Despite Naut Jungle being my bread and butter, I've taken him bot too as of late.) Still, don't be afraid to experiment or do new things. I've had success with some strange builds like AP tank Lissandra. You won't know how far you can extend until you over-extend a bunch. You'd be surprised the kind of hooks I can hit simply cause I've just got it so burned in my head. I always find it hilarious to scoop up dudes with Naut that my team didn't even think was in range. If Veigar is your boy stand by him, he's certainly not the worst mid pick by any means!

Few things I would love to say in regards to your post to Hatsune:

Kassadin just snowballed via roaming & killing other lanes. I spammed MIA ping until I couldn't anymore but my ADC still proceeded to find a way to flame me by the end of it all.

Veigar isn't the best roamer, so following can be risky, especially vs a mobile assassin. The best thing to do there would be to spam ping mia like you said, but also ping it on the side lines. Ping mia in mid 2-3 times, then mia on top of the adc the rest of the remaining you can. Try to punish the fact that he is roaming by shoving under tower and getting some chip damage, forcing him to miss cs, or even roam top lane if you can.

Afterwards, I tried to play aggressively so that something like that wouldn't happen again. However, this led to me dying in teamfights due to getting caught in overly aggressive positions.

This is a noble mindset, but ultimately very risky. You are completely altering how you play a champion that you are trying to master. You need to go into your games with a game plan. Are you a passive player? Sit back and farm and try to just do occasional harass. Realize you aren't going to contribute much and set up as many good wards as you can by pushing out the wave and warding the enemy jungle. This way you can still contribute even when you don't feel comfortable following. Are you an aggressive player? Maybe it is worth missing the first few CS to absolutely shut down the enemy mid laner ALA Faker in MSI on cass vs LB https://www.youtube.com/watch?v=oM8UHxR46uY

I don't want to put full blame on my team because I know I could've carried harder

This is the perfect mentality to have. Keep this thinking and you will climb.

I've been becoming more adequate at exercising my basic game knowledge surrounding facts about items, rotating around the map, awareness of the map, champion strengths/weaknesses, etc.

This is something that comes with time, but only if you actively practice it. Say to yourself in one game "Ok, I don't care if I win this normal draft (because it emulates ranked draft), I am going to focus on getting deep vision of the jungler so that I can stand on the opposite side of the lane, and let my side lanes know where he is!" as an example. You can do this with any basic mechanic you want to learn. The easiest one is learning how to cs better. Set up the training tool, and with no items at all, practice CSing for a couple hours until you can consistently get 80-90 cs at 10 minutes. That should go a long way in your practice with Veigar. Yes it will be practice for playing against no one, but it will still help show you things you need to know.

I'm on the verge of quitting Veigar, but is this the right thing to do? There are plenty of benefits from mastering and playing one champion & I love playing him but it feels like he won't get me anywhere at this rate.

This is kind of a loaded question. On one hand, you aren't winning on the champion, so it can seem like he doesn't fit your playstyle. On the other, since you love him, if you master him it will go much further than if you master a meta pick that is going to get nerfed repeatedly. If you are familiar with Gbay99, he recently posted a video talking about how he was one tricking irelia and got to mid diamond 3 with her. He realized that while he was a diamond 3 Irelia player, he was not a diamond 3 player. He started to widen his champion pool and is stuck in D5 right now, using irelia only when he is about to demote. This is teaching him other tactics for winning the game, as well as making him an over all better player.

This of course may not be the right fit for you either, only you know what is best. However, if you really love veigar, but aren't feeling like you are having much success, maybe try playing him differently. Top tank veigar is super off meta, but because of the ap acquisiton and high number of melee, you can get basically a free lane to farm in with Q and harass, allowing you stack up some AP and become a bruiser. Support veigar as an ap or a tank also works because your cage is very good in the long 2v2 lane as it lets your jungler have more time to get to them even if they don't get stunned, and is one of the few abilities that can multi stun at level 3.

Ultimately, I would say don't ONE TRICK Veigar, but main him. Keep a few other champions in your pocket as well though. Figure out your playstyle and add champions that compliment that. My team's mid laner was struggling because he could only play Zed, Syndra, and LB. So I listed a bunch of champions that could fit what he likes to do as a player and he settled on learning because he is the type who likes to win by roaming a lot and maybe winning laning if he can. Find out your playstyle and learn more champs who do just that. Similar champs to veigar if you are looking for the "win lane, win game" style champs... are all champs who don't have to roam to feel effective. If you want more of the "I can farm and carry late" are good options, and if you are thinking "I need more impact on the map" could be great. Ultimately it is up to you to decide how you need to move forward as a player and with your champion pool.