Just roll a d1000 in anydice.

The probabilities for rolling 3d10 as the 3 tens places will be exactly the same as rolling a d1000.

These answers shows the math for d100 vs 2d10, it's exactly the same story here just times ten.

The point of using d1000 is that probabilities are easy to calculate: the chance of the number or less is equal to the number in per-mille. The chance of rolling less than 900 is 900-1 in 1000 or 899‰ (89.9%); the chance of rolling equal or higher is the remainder in 1000 or 101‰ (10.1%) _{SevenSidedDie}

PCs are exceptional regardless of their ability scores

Answering your actual question rather than the math one first: Your hang-up is that you place too much emphasis on ability scores. A fighter is already exceptional just for being a fighter, just as a wizard, cleric or druid are already exceptional for belonging to these classes. Ability scores influence things the characters can do, but that's at best describing the relative strengths of PCs against each other - and even then it's a very inexact and misleading measure, my dwarf barbarian with 28 strength is not really more exceptional than sorcerer with their highest stat being a 17 in charisma, but with the ability to destroy castles at a whim via Earthquake.

Thus, I don't believe your problem can be solved via math, certainly not by slightly tweaking the ability scores players get; It's more a matter of framing PCs as larger than life heroes, which is very much within your purview as a DM.

The literal math question

Here are the exact (to six significant digits) probabilities of both getting the individual scores and getting a score that is at least as great as a particular score (using the 4d6 drop lowest system). But like I said earlier, I'm not sure that knowing this will improve matters much for you (note that an "average" human is assumed to have values at around 10 (see also the Commoner Statblock):

Individual Score Probabilities

\$ \begin{array}{c|c|c} \text{Ability Score} & \text{Probability} & {\text{Probability of being}\\\text{at least as great as}}\\\hline 18 & 1.620370\% & 1.620370\%\\ 17 & 4.166667\% & 5.787037\%\\ 16 & 7.253086\% & 13.040123\%\\ 15 & 10.108025\% & 23.148148\%\\ 14 & 12.345679\% & 35.493827\%\\ 13 & 13.271605\% & 48.765432\%\\ 12 & 12.885802\% & 61.651235\%\\ 11 & 11.419753\% & 73.070988\%\\ 10 & 9.413580\% & 82.484568\%\\ 9 & 7.021605\% & 89.506173\%\\ 8 & 4.783951\% & 94.290123\%\\ 7 & 2.932099\% & 97.222222\%\\ 6 & 1.620370\% & 98.842593\%\\ 5 & 0.771605\% & 99.614198\%\\ 4 & 0.308642\% & 99.922840\%\\ 3 & 0.077160\% & 100.000000\%\\ \end{array} \$

Expected single ability score value: 12.24

Expected total: 73.47 (as you suspected in the question)

And here are the probabilities for score totals (up to 6 significant digits). Note that there are a bunch of zeros and 100%s in here, these are mostly because the chances are just so astronomically small or big that 6 significant digits aren't enough (except for the accumulated probabilities for minimum values, since of course nothing can be smaller than the minimum).

Total score probabilites

\$ \begin{array}{c|c|c} \text{Total Score} & \text{Probability} & {\text{Probability of being}\\\text{at least as great as}}\\\hline 108 & 0.000000\% & 0.000000\%\\ 107 & 0.000000\% & 0.000000\%\\ 106 & 0.000000\% & 0.000000\%\\ 105 & 0.000001\% & 0.000002\%\\ 104 & 0.000006\% & 0.000007\%\\ 103 & 0.000022\% & 0.000030\%\\ 102 & 0.000073\% & 0.000102\%\\ 101 & 0.000212\% & 0.000314\%\\ 100 & 0.000559\% & 0.000873\%\\ 99 & 0.001355\% & 0.002228\%\\ 98 & 0.003053\% & 0.005280\%\\ 97 & 0.006438\% & 0.011719\%\\ 96 & 0.012800\% & 0.024518\%\\ 95 & 0.024120\% & 0.048638\%\\ 94 & 0.043277\% & 0.091916\%\\ 93 & 0.074221\% & 0.166137\%\\ 92 & 0.122070\% & 0.288207\%\\ 91 & 0.193078\% & 0.481284\%\\ 90 & 0.294415\% & 0.775699\%\\ 89 & 0.433730\% & 1.209429\%\\ 88 & 0.618481\% & 1.827910\%\\ 87 & 0.855065\% & 2.682974\%\\ 86 & 1.147820\% & 3.830794\%\\ 85 & 1.498018\% & 5.328812\%\\ 84 & 1.902979\% & 7.231791\%\\ 83 & 2.355468\% & 9.587259\%\\ 82 & 2.843496\% & 12.430755\%\\ 81 & 3.350615\% & 15.781370\%\\ 80 & 3.856750\% & 19.638120\%\\ 79 & 4.339492\% & 23.977612\%\\ 78 & 4.775768\% & 28.753380\%\\ 77 & 5.143685\% & 33.897065\%\\ 76 & 5.424356\% & 39.321421\%\\ 75 & 5.603493\% & 44.924914\%\\ 74 & 5.672574\% & 50.597488\%\\ 73 & 5.629472\% & 56.226960\%\\ 72 & 5.478471\% & 61.705431\%\\ 71 & 5.229710\% & 66.935141\%\\ 70 & 4.898120\% & 71.833262\%\\ 69 & 4.502022\% & 76.335284\%\\ 68 & 4.061548\% & 80.396832\%\\ 67 & 3.597061\% & 83.993893\%\\ 66 & 3.127736\% & 87.121629\%\\ 65 & 2.670427\% & 89.792056\%\\ 64 & 2.238874\% & 92.030931\%\\ 63 & 1.843300\% & 93.874230\%\\ 62 & 1.490344\% & 95.364575\%\\ 61 & 1.183308\% & 96.547883\%\\ 60 & 0.922602\% & 97.470485\%\\ 59 & 0.706332\% & 98.176817\%\\ 58 & 0.530938\% & 98.707756\%\\ 57 & 0.391803\% & 99.099559\%\\ 56 & 0.283804\% & 99.383363\%\\ 55 & 0.201753\% & 99.585116\%\\ 54 & 0.140730\% & 99.725846\%\\ 53 & 0.096296\% & 99.822142\%\\ 52 & 0.064621\% & 99.886763\%\\ 51 & 0.042516\% & 99.929279\%\\ 50 & 0.027414\% & 99.956693\%\\ 49 & 0.017318\% & 99.974011\%\\ 48 & 0.010713\% & 99.984724\%\\ 47 & 0.006487\% & 99.991211\%\\ 46 & 0.003842\% & 99.995053\%\\ 45 & 0.002225\% & 99.997279\%\\ 44 & 0.001259\% & 99.998538\%\\ 43 & 0.000696\% & 99.999233\%\\ 42 & 0.000375\% & 99.999608\%\\ 41 & 0.000197\% & 99.999805\%\\ 40 & 0.000101\% & 99.999905\%\\ 39 & 0.000050\% & 99.999956\%\\ 38 & 0.000024\% & 99.999980\%\\ 37 & 0.000011\% & 99.999991\%\\ 36 & 0.000005\% & 99.999996\%\\ 35 & 0.000002\% & 99.999998\%\\ 34 & 0.000001\% & 99.999999\%\\ 33 & 0.000000\% & 100.000000\%\\ 32 & 0.000000\% & 100.000000\%\\ 31 & 0.000000\% & 100.000000\%\\ 30 & 0.000000\% & 100.000000\%\\ 29 & 0.000000\% & 100.000000\%\\ 28 & 0.000000\% & 100.000000\%\\ 27 & 0.000000\% & 100.000000\%\\ 26 & 0.000000\% & 100.000000\%\\ 25 & 0.000000\% & 100.000000\%\\ 24 & 0.000000\% & 100.000000\%\\ 23 & 0.000000\% & 100.000000\%\\ 22 & 0.000000\% & 100.000000\%\\ 21 & 0.000000\% & 100.000000\%\\ 20 & 0.000000\% & 100.000000\%\\ 19 & 0.000000\% & 100.000000\%\\ 18 & 0.000000\% & 100.000000\%\\ \end{array} \$

Source code for generating the table (Haskell)

{-# LANGUAGE Strict, TypeApplications #-}
import qualified Data.Map as M
import Text.Printf
import Data.List(sortBy)
import Data.Ord(comparing)
import Control.Applicative((<|>))
import Data.Ratio

rolls :: [Int]
rolls = do
  a <- dice
  b <- dice
  c <- dice
  d <- dice
  return . sum . take 3 . sortBy (comparing negate) $ [a,b,c,d] where
  dice = [1..6]

frequencies :: M.Map Int Int
frequencies = foldr (\v m -> M.alter (\f -> fmap (+1) f <|> Just 1) v m) M.empty rolls

probabilities :: M.Map Int Rational
probabilities = fmap ((/ 6^4) . fromIntegral) frequencies

printProb :: Int -> Rational -> Rational -> IO ()
printProb k p acc = printf "%d & %.6f\\%% & %.6f\\%%\\\\\n" k (percent p) (percent acc) where
  percent = fromRational @Double . (100*)

totalScoreInstances :: [(Int, Rational)]
totalScoreInstances = do
  (strength, pStr) <- oneScore
  (constitution, pCon) <- oneScore
  (dexterity, pDex) <- oneScore
  (wisdom, pWis) <- oneScore
  (intelligence, pInt) <- oneScore
  (charisma, pCha) <- oneScore
  let totalScore
        = strength
        + constitution
        + dexterity
        + wisdom
        + intelligence
        + charisma
  let totalProb
        = pStr
        * pCon
        * pDex
        * pWis
        * pInt
        * pCha
  return (totalScore, totalProb)
  where oneScore = M.toList probabilities

totalScoreProbabilities :: M.Map Int Rational
totalScoreProbabilities = foldl
  (\m (s, p) -> M.alter (\tp -> fmap (+ p) tp <|> Just p) s m)
  M.empty
  totalScoreInstances 

main = do
  putStrLn "## Individual Score Probabilities"
  putStrLn "\\$"
  putStrLn "\\begin{array}{c|c|c}"
  putStrLn "\\text{Ability Score} & \\text{Probability} & \\text{Probability of being at least as great as}\\\\\\hline"
  fst $ M.foldrWithKey (\k v (m,acc)  -> let acc' = acc+v in (m >> printProb k v acc', acc')) (return (), 0) probabilities
  putStrLn "\\end{array}"
  putStrLn "\\$"
  let expected = M.foldlWithKey (\t k v -> t + (fromIntegral k * v)) 0.0 probabilities
  printf "Expected single ability score value: %.2f\n" $ fromRational @Double expected
  printf "\nExpected total: %.2f\n" $ fromRational @Double expected * 6
  putStrLn "## Total score probabilites"
  putStrLn "\\$"
  putStrLn "\\begin{array}{c|c|c}"
  putStrLn "\\text{Total Score} & \\text{Probability} & \\text{Probability of being at least as great as}\\\\\\hline"
  fst $ M.foldrWithKey (\k v (m,acc) -> let acc' = acc+v in (m >> printProb k v acc', acc')) (return (), 0) totalScoreProbabilities
  putStrLn "\\end{array}"
  putStrLn "\\$"

Note: if you want to run this yourself, compile with ghc -O2 and don't try to run it via the interpreter if you want to see the result before the heat death of the universe.