LSU went the better part of 50 years without said advantage. Their "rise" is a relatively new phenomenon. Up until that midget cocksucker got there LSU was basically Ole Miss with a better uniform.
If you go by all time winning percentage, that's not true. LSU is 13th in all time winning percentage, Auburn is 15th.
But the real "advantage" is being "The state school"
Top 20 all time in winning percentage.
12 of the top 20 are "the state school", 2 are private, one is "the national school".
No/School/W%/W/L/Tie/Total games played.
1 Michigan 0.73982 872 295 36 1203
2 Notre Dame 0.73639 831 284 42 1157
3 Texas 0.72039 835 314 33 1182
4 Oklahoma 0.71472 789 299 53 1141
5 Ohio State 0.71441 807 307 52 1166
6 Alabama 0.70855 799 316 43 1158
7 Southern Cal 0.70614 766 303 54 1123
8 Nebraska 0.70275 824 337 40 1201
9 Tennessee 0.69214 783 333 55 1171
10 Penn State 0.68934 800 349 42 1191
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11 Florida State 0.67073 459 221 17 697
12 Georgia 0.64599 723 384 54 1161
13 Louisiana State 0.64027 700 383 47 1130
14 Miami-Florida 0.63402 544 310 19 873
15 Auburn 0.62734 681 395 47 1123
16 Florida 0.62713 641 373 40 1054
17 Miami-Ohio 0.62593 649 379 44 1072
18 Washington 0.61601 652 397 50 1099
19 Arizona State 0.61500 541 334 25 900
20 Central Michigan 0.60999 557 350 34 941