Any AP heroes since those will be cheaper on rune costs i'd imagine due to me having mpen/ap runes already? That's part of the reason I was thinking GP and his parrrleys.
Since you seem to be starting to ask relevant questions to getting better rather then just making excuses.. I'll give you a real answer and say the first things you need to buy are flat armor yellows and scaling MR/lvl blues. You will use these on a majority of champions, regardless of role. If you simply made a page with those, and then just made one AD page with flat AD reds and quints and an AP page with flat AP reds and flat AP quints (or arpen/mpen quints since they are slightly more effective then arpen/mpen reds).. you'd at least be close enough to optimal in most matches. Until you have a bunch of rune pages, that kind of setup is 'good enough' for most champions and situations.
Having the right runes is more cost effective then buying champs, as you will be able to make an optimal page(s) for whatever champion you play, which is more important than having a 'counter pick' champion available that you barely ever play. The advantage in having correct runes/masteries is higher then playing a counter pick champion if they aren't one of your 5 most played champs.
But, as others said, an AD based bruiser champ is the best bet. It's less reliant on your teammates for you to win lane and take a tower to give them an advantage, then it is to win lane and immediately try to gank for them if your timing isn't on point. Plus, when you take your lane's first tower, you get more roam time when your lane pushes up so you can then spend more time trying to setup ganks.
On the same note, some AP champions, especially ones that build a lichbane can push towers pretty well. Ziggs is amazingly strong at taking towers if you get a lichbane combined with his passive (unless they changed his passive working on buildings, haven't played him in awhile) as is Twisted Fate if you build lichbane. They both clear waves quickly and then do better damage to towers then most AP champions because they make use of the lichbane proc better. Or, of the champions you listed, anivia and morgana are great at clearing waves which pushes the lane with minions, but slower than an AD based champ. Veigar and Annie are too fragile to hard carry a game easily, and kassadin is shitty early game. It's hard to make a big impact early with him, he's best mid-late game when he can start assassinating people in teamfights and escaping.
I'd echo the previous suggestion of Ryze, as he builds pretty tanky even in his max damage build, clears waves pretty well and duels great due to his natural item progression and good sustained damage output. Or, (rough order of strength for early game) kennen, rumble, diana, gragas and vlad are all better overall choices to pubstomp then the others you mentioned. They can clear waves quickly and do well early game if played correctly. Vlad isn't great till lvl 7+ for example, so he's the weakest of those choices, and I personally enjoy playing him alot. Just saying he rarely gets kills on his own till that point, and you want to make an impact before that point to get the game snowballing in your team's direction. Gragas only if you get some good poke down, he can then spike the remaining 50% of their health or at least force them out of lane so you get a level advantage plus he has good sustain naturally so you can sit in lane longer for more of a level advantage.
Early on, levels mean as much, or more, than items. A 2 level advantage in the first 6 levels is like having a big item just in the extra ranks of skills. A good thing to remember when someone gets away with a sliver of health at lvl 3 and you are debating tower diving them to get the kill. Getting them to leave lane while you CS a full wave and push it into their tower to deny them, gets you gold/exp that they aren't getting. It makes the next fight go even more in your favor, which leads to snowballing.