I used to main Sivir, and I totally wrecked with her. She had the sustained damage to compete with most other ADCs, and could even be played as a mid-lane burst damage champion.
It's been harder lately, because I think she's been subtly nerfed, while other ADCs have mostly been getting buffs. Spellshield seems kind of bugged, alot of time I activate it and it does nothing, even though visually it looks like it should have worked (and I've improved my internet connection so ping is almost never a problem for me anymore). I also think in the current metagame her ultimate needs a significant buff, either letting the speed boost last longer and fall off more slowly, or (my personal preference) make effected targets immune to slowing for the duration of the effect.
Also, reduced cooldown would be nice for an ultimate that is basically pure utility.
As for how to succeed with her in the modern game, I've struggled, but I still pull out wins with her from time to time. Her wave clear is very solid, and she's great at pushing waves even at early levels. I like to start with W instead of Q in order to help her push, plus it costs less mana (and that's a major issue early on), and I don't pick up Q until level 3, but then I max it first.
One thing most people don't consider is that Sivir's ultimate has a passive that makes her W an AS steroid. As such, it's often a great idea, especially in the late game, to pop W whenever attacking a turret. Hopefully your team's tank will be grabbing aggro from the turret, if there are champions around, otherwise you'll just help to clear the minions while attacking the turret and benefiting from that improved AS.
The biggest buff they've given her recently was possibly unintended, in the form of Essence Reaver. She truly excels with this item, particularly now that it matches BT for AD bonus. Some people might say she doesn't need it, due to her E, but E is a stop-gap/defensive measure, and even if you can regularly get it off, ER still grants some great bonuses for her.
CDR is very powerful on Sivir, and as such I like to build not only Essence Reaver, but also Ionian Boots of Lucidity, and Zephyr (usually my last item). She also suffers from not having any CC, so I like to build a Frozen Mallet immediately after rushing Essence Reaver. You might not need to if you have really strong CC on your support, but I've always done well with it. Besides it really fits the theme of my Snowstorm Sivir (though I call her Shivir).
For my remaining two items I like Infinity Edge and Last Whisper. Both fairly standard on ADCs. I also like the Alacrity enchant on my boots.
Hope some of this helps.