I played champions with strong mid-game teamfighting and either 1) good poke and waveclear or 2) strong snowballing potential early- and mid-game. In addition, I recall that many players in Silver would pick champions with high damage, low durability, and low CC meaning that champions with durability and/or CC were very useful for covering the team's weaknesses.
My main picks in Silver (my first ~30 ranked games if you want to check) were:
ADC
Mid
Jungle
Top.
They worked quite well for me.
Both teams would often group around an objective (i.e. tower, Dragon, Baron) and proceed to stand around lobbing pot-shots are one another until someone finally engaged (for better or worse) during these stand-offs.
Ziggs and Varus were both capable of waveclearing and poking down the enemy team without any risk to themselves (and Silver players aren't the most competent at dodging skillshots). When either team eventually engages, their AoE CC and damage would wreak havok with the enemy team.
Gragas could help waveclear and poke (Q) while enduring enemy harass (passive). However, his ultimate is the most useful in this scenario since it can be used to disengage fights, pick off enemies, or split up the enemy team. Even without his ultimate, his Q and E provide good CC and his W is a great defensive steroid.
Nasus was a very reliable pick for me in lower ELOs. Many players don't know how to properly pressure a lane, coordinate ganks, kite, or handled splitpushing which means that it's quite easy to farm, pressure, and teamfight as Nasus. They don't really know how to properly control a minion wave or break a freeze either, which can let you farm safely under your tower while the enemy is constantly overextended. His W is a powerful debuff and CC, his E provides decent waveclear and armor reduction, his sustain scales rather quickly (especially if you have 400 stacks and Sheen by 20 minutes, which isn't that hard at low ELOs), and he's difficult to tower-dive (especially if you don't know how).
Honestly, if you really don't belong in Silver you should get out pretty quickly. You can carry in any role when you play it properly; some are just more obvious than others.
Taking a quick, quick look at the game you linked, I would suggest you work on your warding. You had the yellow trinket for the entire game but only placed 3 wards.
Also, you shouldn't have built Mortal Reminder that game. The only champion that would have been useful against was Thresh (since his passive gives him bonus armor). Trundle had no armor, Vladimir only had Zhonya's Hourglass and Thresh had a Cloth Armor; if you bought it to counter their sustain, you should have left it at Executioner's Calling until it was your last item to upgrade. You'd have done better with Bloodthister instead of it.