This is so stupid I don't even know where to begin.
First, throw away the two pieces of anecdotal evidence, they have been are and always will be worthless.
Second, the two people in you ending analogy are equal because you've made them equal by giving the second person a non disclosed amount of time to get the same materials as the person who got them much quicker through buying.
So here, have a more in depth hypothetical scenario.
Two people start playing LoL at the exact same time. These two people are exactly equal in skill.
One person is given an unlimited amount of money to spend on champs, ip boosts and xp boosts. The other is given no money at all.
They are only allowed to play for the same amount of time per day per week, and after 1 week they will 1v1.
Also assume that at the end of one week they are still the same level of skill, and any amount of outside research was done equally as well.
One player will have access to more (in this theoretical case all) of the champs, and will thus have been able to use all of their earned ip (of which they have more of from the boosts) on runes. They will also have access to more masteries and summoner spells, unless you want to have both be able to level to 30, which doesn't really change much.
The other person will have whatever combination of champs and runes they were able to afford with the ip gained.
Which will win? Without making guesses, the first person has a clear advantage over the free account player.