what would a confernece comprised of each states flagship athletic department look like?
In order to determine the flagship athletic department for each state I collected NCAA tournament results from each sport over the last 20 years 1998/99-2017/18. This data is available on ncca.org & wikipedia.
NOTE: for football the BCS top 25 final ranking and then then playoff final top 25 ranking was used.
NOTE: Did not include the NIT or any other non-NCAA tournament results.
NOTE: any school that participated in any NCAA sponsored tournament is included even if they play the majority of their sports in Division II. Alaska Fairbanks is an example of a school that is primarily division II but won 9 division I national championships.
A first place finish/national championship in any sport was awarded with 100pts, 2nd place with 99pts and so on. Some tournaments only have 4 teams while basketball has 68 (1st rd losers get 35pts).
All sports are scored the same, so a football national championship or a water polo national championship in any year are both worth 100 points.
All men and women sports are included in the year they are sponsored by the NCAA. Most sports were included each of the last 20 years, but there were a few new sports introduced like Beach Volleyball from 2015-2017.

The result was this list of the flagship athletic department for each state (and DC). The first number is the overall rank including all schools that participated in a NCAA tournament over the last 20 years.Ex.) Stanford ranked #1 easily and was the top school of the 26 California schools that played in at least 1 NCAA tournament over the last 20 seasons:

1 Stanford CA 01 26 #N=42 #T=487 SC=44005 Pac 12
2 North Carolina NC 01 20 #N=17 #T=402 SC=32764 ACC
3 Michigan MI 01 07 #N=7 #T=371 SC=31568 Big Ten
5 Ohio State OH 01 13 #N=11 #T=378 SC=31154 Big Ten
6 Florida FL 01 14 #N=23 #T=340 SC=30325 SEC
7 Texas TX 01 26 #N=16 #T=330 SC=28646 Big 12
8 Penn State PA 01 14 #N=26 #T=339 SC=28552 Big Ten
9 Notre Dame IN 01 09 #N=10 #T=335 SC=26881 ACC
10 Virginia VA 01 16 #N=14 #T=318 SC=26679 ACC
13 Minnesota MN 01 02 #N=12 #T=309 SC=25552 Big Ten
16 Georgia GA 01 08 #N=23 #T=282 SC=24424 SEC
17 Arizona State AZ 01 04 #N=8 #T=297 SC=24401 Pac 12
18 LSU LA 01 16 #N=13 #T=278 SC=23749 SEC
20 Washington WA 01 05 #N=6 #T=276 SC=22787 Pac 12
22 Wisconsin WI 01 04 #N=8 #T=269 SC=21984 Big Ten
23 Tennessee TN 01 13 #N=6 #T=266 SC=21885 SEC
24 Nebraska NE 01 03 #N=9 #T=257 SC=21485 Big Ten
25 Oklahoma OK 01 04 #N=17 #T=252 SC=20749 Big 12
26 Arkansas AR 01 05 #N=15 #T=244 SC=20566 SEC
27 Auburn AL 01 12 #N=13 #T=248 SC=20368 SEC
* 28 BYU UT 01 06 #N=7 #T=258 SC=19988 WCC
29 Kentucky KY 01 08 #N=3 #T=246 SC=19695 SEC
30 South Carolina SC 01 11 #N=4 #T=240 SC=19192 SEC
** 32 Princeton NJ 01 09 #N=5 #T=243 SC=18916 Ivy League
34 Illinois IL 01 17 #N=2 #T=227 SC=18248 Big Ten
35 Oregon OR 01 04 #N=20 #T=209 SC=17468 Pac 12
40 Maryland MD 01 11 #N=18 #T=198 SC=16211 Big Ten
41 Missouri MO 01 08 #N=0 #T=214 SC=15661 SEC
** 43 Harvard MA 01 07 #N=2 #T=198 SC=15107 Ivy League
47 Colorado CO 01 05 #N=12 #T=179 SC=14047 Pac 12
48 Iowa IA 01 04 #N=5 #T=180 SC=13832 Big Ten
* 55 New Mexico NM 01 02 #N=3 #T=152 SC=11378 Mountain West
57 Mississippi MS 01 05 #N=0 #T=151 SC=11138 SEC
58 West Virginia WV 01 02 #N=6 #T=140 SC=11011 Big 12
64 Syracuse NY 01 26 #N=8 #T=126 SC=10466 ACC
65 Kansas KS 01 03 #N=2 #T=138 SC=10439 Big 12
* 66 UConn CT 01 07 #N=18 #T=118 SC=9974 American
** 74 Dartmouth NH 01 02 #N=1 #T=107 SC=8649 Ivy League
* 78 Boise State ID 01 03 #N=0 #T=109 SC=7952 Mountain West
** 84 Providence RI 01 05 #N=2 #T=91 SC=7045 Big East
* 93 UNLV NV 01 02 #N=0 #T=96 SC=6567 Mountain West
* 94 Hawaii HI 01 01 #N=1 #T=81 SC=6548 Big West
**119 American DC 01 03 #N=0 #T=64 SC=4679 Patriot
*151 Wyoming WY 01 01 #N=0 #T=51 SC=3212 Mountain West
**162 Vermont VT 01 01 #N=1 #T=33 SC=2905 America East
**183 North Dakota State ND 01 02 #N=0 #T=43 SC=2577 Summit
**184 Montana State MT 01 02 #N=0 #T=33 SC=2549 Big Sky
**194 Delaware DE 01 02 #N=1 #T=30 SC=2306 CAA
**219 South Dakota State SD 01 02 #N=0 #T=30 SC=1862 Summit
**242 Maine ME 01 01 #N=1 #T=19 SC=1459 America East
**291 Alas. Fairbanks AK 01 01 #N=9 #T=9 SC=900 Division II

Legend :
* : school is in FBS conference but not in a P5 conference)
** : school is not in FBS conference)
#N= : number of national championships won in last 20 years
#T= : number of teams that played in a NCAA tournament
SC= : total score

P5 schools were the flagship athletic department in 32 / 51 States+DC. There are 2 states where a P5 conferece school was not the flagship athletic department of their state: NJ(Princeton) & UT(BYU)

SEC flagship: FL(Florida), GA(Georgia), LA(LSU), TN(Tennessee), AR(Arkansas), AL(Auburn), KY(Kentucky), SC(South Carolina), MO(Missouri), MS(Mississippi)
not flagship: TX(Texas)

B1G flagship: MI(Michigan), OH(Ohio State), PA(Penn State), MN(Minnesota), WI(Wisconsin), NE(Nebraska), IL(Illinois), MD(Maryland), IA(Iowa)
not flagship: NJ(Princeton) IN(Notre Dame)

P12 flagship: CA(Stanford), AZ(Arizona State). WA(Washington). OR(Oregon), CO(Colorado)
not flagship: UT(BYU)

B12 flagship: TX(Texas), OK(Oklahoma), WV(West Virginia), KS(Kansas)
not flagship: IA(Iowa)

ACC flagship: NC (North Carolina), IN(Notre Dame), VA(Virginia), NY(Syracuse)
not flagship: SC(South Carolina), KY(Kentucky), FL(Florida), GA(Georgia), MA(Harvard), PA(Penn State)