Malphite is the Hard Engage of League. He is raw tank and his Ultimate is just him busting into a team fight like the Kool Aid Man, knocking everyone up at the target location. Can't get much of a harder engage than that. He works extremely well with follow-ups such as Yasuo or AoE ultimates like MF. He scales with armor and so can easily build up to counter the enemy ADC and top (and jungle.. and mid... these days its pretty heavy AD comps). He has some decent lane harass and manageable clearing. His passive (gives him a %hp shield after not taking damage for a few seconds) makes him resistant to poke in the laning phase. He is generally played top, as tank jungles are suffocating right now, but I've also seen him as a support in off-meta comps.
As you've found, the "Hard Engages" are usually on champions who are durable and tanky, but if you want a mage that can engage, I recommend Kennen or Fiddlesticks. Both of them are high-damage, relatively squishy mages who are basically built to get a Zhonya's Hourglass and then charge head first into a fight. Both of them have AoE Ultimates that are not channels so they can jump in, ult, and then go into Zhonya's. Fiddle is a beast jungler and his massive sustain via his drain means he doesn't have to recall until he wants to, while Kennen has a somewhat outdated and clunky feeling laning phase. However Kennen's ult can stun multiple champions so if played correctly he can easily set up his team. Of the two, Fiddle is more "I'M GOING IN GUYS" because you basically sit behind a wall and then ult into the enemy team.
Typically ADC's aren't necessarily the ones to initiate a fight, but if you are looking for an ADC that can say "Teamfight, go!" then you want Ashe. Even though her kit is very simplistic and there is nothing fancy about her, her ultimate is a skillshot global stun so she can start a fight anytime her ult is up.
(best)
(pre nerf)
mid
adc/top
top
jung
sup
adc
mid/jung
has stupidly good single target engage with his W and R starting at level 6. Is also a 1350 ip dude. ..... Late game you are basically going to be 1 v 2ing or 1 v 3ing people while soaking up a ton of damage.
land a hook, start a fight
he's got like what 8 forms of cc? hows that for engage
yasuo's beastial butt buddy
in sec
drunk in sec
yolo in sec