**I am now %100 convinced that the corners first method is better for fewest moves, less algorithms to memorize, shorter algorithms, easier recognition, and most likely faster times.**

after all, think about this, there are only two types of unique corner pieces. like binary, each corner is correct or it is not correct. the maximum number of moves to permute (but not care to orient) one corner piece from a scrambled cube is 1 move in QTM (with one exception). there is however a very un-lucky rare case of having all 8 corners in a specific configuration that makes this statement untrue slightly. in %1.42857142857 of scrambles (4/8*3/7*2/6*1/5) you will have for example corner type A in LFU RFU RFB RFD also with corner type B in LBU LFD LBD RBD. given that in this example that RFU should contain type B as the correct corner piece, for this special case you can not move the correct corner piece into the RFU position with 1 move in QTM because LFU RFB RFD all contain type A pieces. a type B corner piece must be moved from one of the LBU LFD LBD RBD positions to the RFU position using 2 moves in QTM. also 3 moves QTM if the corner piece is taken from LBD to be placed in RFU.

on average there will be 4 corners not permuted. I do not know if the distribution is flat or gaussian. I do know that if you plan your moves carefully that you can position 1, 2, 3, or 4 corners pieces simultaneously using 1 move in QTM. the lower bound for move length in QTM to place but not orient all 8 corners from any given scramble would likely be based on the lower bound to generate the worst case scenario given in the example above from a solved cube. I can manually try the upper bound by hand. we know it is less than or equal to the 14 move superflip provided in this thread. I do not have proof but I think given the 1 move QTM solutions to place any given corner it would certainly be around 8 plus or minus a few.

in conclusion I think that a corners first solver can orient and permute all the edges after permuting and orienting all the corners with a combination of slice moves and 180 face turns (L2 R2 U2 D2 F2 B2) that will preserve the appearance of corner sticker colors. there are indeed even more algorithms that do not conform to this convention but do preserve the appearance of corner sticker colors. it is likely that none of these algorithms are greater than 7 moves unless we have a system of solving all 12 edges at once. a 32 move speed solve is reasonably easy to achieve using this method. that number will be reduced by combining edge orientation and edge permutation into a one look. we can also use edge influence when solving corners intuitively since most corners first solves of a 3 color cube have many equally efficient alternate mirrors. that would bring a speed solve down to 24 moves.