I'd say your way of assessing greatest is severely flawed!
First off, level of opposition. Morphy would have gotten creamed if anybody that was of any strength at all existed then.
Secondly, Karpov would have creamed Fischer
Thirdly, as an off-shoot from the first, length of time means nothing because again, it's a matter of timing of when another strong player comes into the picture. Take American Football. If one team in the division goes 11 and 5, and nobody else in the division wins more than 6 games, and this happens for 8 years straight, is this the greatest team ever because they won their division 8 years in a row? Uhm - NO! It's because their opposition at the time was weak!
Therefore, the top 5 list should look something more like this:
1) Kasparov
2) Carlsen
3) Karpov
4) Anand
5) Kramnik
Also, on your flawed system, Botvinnik was WC for 13 total years, and actually played all 13 years, not like that stupid clown that was world champion for 3 years and then refused to play Karpov, and sat idle during those 10 years that you claim was his! Karpov's brain does not work as a proxy for Fischer!
Who are the 5 greatest chess players of all time according to you and why?
My view:
According to me, the 5 greatest chess players of all time are:
1) Paul Morphy.
2) Emanuel Lasker (27 yrs)
3) Alexander Alkhine (17 yrs)
4) Garry Kasparov (15 yrs)
5) Fischer
I have simple way of judging who is the greatest of all time. If a player has reigned as the world champion for the longest period, then he is the greatest in my view. The only other factor that might influence this metric would be if a player has voluntorily stopped playing chess for some reason and may still be arguably the best in the world in that time period. And there are only two such cases: Paul Morphy & Fischer. In these cases, their reign period would be until the rise of a new champion. Paul Morphy played in 1850s and retired in 1860s(even though he was still the best). Then, Steinitz was there but its unclear when he should be treated as a champion. I am putting Steinitz champion from around 1880s. So, according to me, Morphy remained undisputed champion for 30 yrs(1850s - 1880s) even though he had stopped playing chess in 1860s. Similarly, Fischer voluntorily pulled out and Karpov became the nominal champ. So, Karpov was second-best and his reign also belongs to Fischer. That would mean Fischer reigned for around 13 yrs effectively.