In the description of the sentinel shield it says "While holding this shield..." (DMG page 199) so you must be holding the shield, not just have it on your back. From a quick glance, many other shields have this condition also.

However, you CAN still use a shield that you aren't proficient with but "you have disadvantage on any ability check, saving throw, or attack roll that involves Strength or Dexterity, and you can’t cast spells" (PHB page 144)

I wrote a python script to calculate answers accurate to several decimal places. I created a Moonblade object, and added a specified number of runes just as the rules suggest (with the one alteration you made that you always re-roll non-stackable properties). It then calculates the expected damage from this moonblade taking into account all bonuses, bonus dice, critical hits, creatures, etc.... I made a couple of assumptions beyond what you have explicitly stated.

1. I assumed that the creature on the receiving end has a uniformly random chance of being any of the 14 types. This is a bad assumption for most games (how often do you see a plant or an ooze compared to a humanoid?), but absent more data, I cannot do better.
2. I assumed that the roll to hit was greater than 1. Though you did say to assume that the hit has been made, in order to calculate the chance of a hit being critical, we need to know the chance of the d20 showing a sufficiently high number given that we know that it is high enough to hit in the first place. The chance of a random d20 roll being a 20 is 1/20, but the chance of it being 20 given that it had to be able to hit an AC 18 creature could be significantly higher. Since we do not have an AC to measure against, the best we can do is say that we know the roll must be higher than 1 (as a natural 1 never hits).
3. I assumed that the user never uses the property of the defender whereby he can transfer some of the sword's hit bonus to his AC.
4. I assumed the user always uses one hand.
5. I assumed the user has a damage modifier of 5 because you would expect this of the high-level character wielding the moonblade.

With these assumptions, I get that with 99% confidence, the true expected damage values given \$n\$ runes are these, \$\pm 0.05\$.

\begin{array}{ll} 0\text{ runes}&9.737\\ 1\text{ runes}&10.244\\ 2\text{ runes}&10.752\\ 3\text{ runes}&11.252\\ 4\text{ runes}&11.727\\ 5\text{ runes}&12.166\\ 6\text{ runes}&12.557\\ 7\text{ runes}&12.899\\ 8\text{ runes}&13.196\\ 9\text{ runes}&13.459\\ \end{array}

This plot demonstrates the almost linear relationship.

For the first 10 or so runes, the expected damage is about \$9.912 + 0.421 n\$. However, the linearity drops off at about \$n=10\$. This is because we expect to have reached a bonus of +3 by this point.

If we extend to a truly absurd number of runes, we see a transition to another linear behavior. I think this is because the dominant factor in the damage is the steadily growing number of d6's added to the roll. Because all other runes eventually stop stacking, in the end, all we can do is add another d6 for each rune, meaning we either add 1d6 to the damage, or add 1d6 to the damage for a particular type of creature. After adjusting to account for critical hits, this averages out to about 2.07 damage per rune, which is what we expect the slope to eventually be. Therefore, for large numbers of runes, the expected damage is calculated as \$\overline{d}\approx-93.24 + 2.07 n\$