Recently, I have been in several long games, where I have a full build. At this point I have to decide whether to keep my boots, and still be fast, or to sell them to open up space for an item that gives better stats. I don't like having to choose, and this got me thinking: Why don't we have a boot trinket slot like we have a ward trinket?
Generally speaking, choice is good.
Also - the average game of LoL is ~33 minutes long and sees ~11,000 gold per player, which means barely enough gold to complete 3 major items and boots - 1 item of which you might just have the money for but inability to complete, seeing as that last team-fight and exploded Nexus turrets/inhibitors can give a solid chunk of gold.
You're confusing most games with excess space most players won't get to use. You're really going to screw with newer players who are trying to make 5 major item builds, giving them a solid 3 item slots they'll probably never use rather than 2, increasing the likelihood they'll kill their build by placing core items in time/gold slots they'll never reach.
Generally speaking, early, middle, and late game can be segmented into major item purchases because of this as well.
5 major items is late-late game and very much favors late-game champions as-is.
Giving the option of 6 major items with no trade-offs to speak of further expands the late-game power of late-game champions, making them even tougher to deal with.
Finally - giving more item slots to players further expands the valley of power between champions that are snowballing ahead or have a significant advantage compared to their opponents, leading to more one-sided games.
Currently, there's at least a small chance you can turtle/play defensively and catch up to the item levels of a team that pulled ahead due to the 6 item slot limit. It can take pushing a game into 50+ minutes, no easy task, but it's just possible enough ATM you've still got some hope of pulling off an upset.
A side boot slot would significantly harm that particular capability of the underdog coming back, leading to more one-sided games.
And one-sided games aren't that interesting to watch or play.
If Riot ever came up with a map that favored really long, drawn-out battles - then yeah, more item slots would be cool. If the average game length was closer to 50 minutes and average income closer to 16,000 gold - then that would be a practical thing to have, seeing as everyone would hit those caps.
In a ~33 minutes, ~11,000 gold game - not so much.