The idea isn't too great, but your logic is good, you just missed a bit in the last part XD
But congratulations - you're among the few new players to notice on their own that saving gold and getting BF sword first will generally be negative for you, as while you're saving you will die/not kill because your oponents would have traded gold for items.
The solution to this problem is to not start with infinity edge/bloodthirster/black cleaver. (those are the only items that need BF sword btw). You need to boost your damage with smaller items first, and choose them good, because in the perfect scenario you won't need to sell them. You can always invest in a long sword. It gives you 10 damage, and can be build into :
-phage (itself build to frozen mallet or trinity force)
-brutalizer (itself build into ghostblade)
-last whisper (good item in general)
-Mardred's Razors (build into bloodrazors - one of the top dmg items)
As you can see, you have options. Get a brutalizer. +25 dmg and +20 armor pen is incredible boost for just 1500, and is build from two long swords. Then you can get an avarice, which will boost your crit, and gold income. Keep it in your inventory for the gold, and when you need the slot merge them to ghostrblade.
There are VERY few champions that can start with infinity edge, and usually this involves geting a bunch of avarice blades first to help you farm the gold. In general, the items that involve BF sword are left for later.
P.S. I said your idea is bad because generally it's not recommended to sell items. You can always plan your build and use itemization routs that will allow you to start small and upgrade later, without having to sell anything.