Personally, I prefer going BT>BotRK>LW instead of BT>LW as it helps a lot when your ultimate is on CD, but I see what you mean. I have played both builds and managed to do well with both, I just prefer BotRK before LW. I'll add in an alternate build going LW before BotRK with a side note.

I don't like rushing IE, it costs a lot and without any other critical chance items, it's a bit lackluster. I have it listed as a 6th item as it does boost damage greatly once you already have crit. Perhaps getting it earlier might benefit more I'd have to check in a couple of games.

Your build sequence is largely suboptimal.

Typically, Sivir wants to go Bloodthirster -> Last Whisper for a lot of damage on Q.

Alternatively, the other typical build is to rush Infinity Edge in order to take advantage of the huge steroid on R.