For the greater good of role play is a fine way to DM, except...
you need to establish that ahead of time.
Look at the Dungeon Master's Guide about dice rolls
There is a chapter in the DMG about running the game, but it's after the chapter on magic items so most people never get that far.
In particular, there is a section called "The Role of Dice". I'll quote a few pieces:
Dice are neutral arbiters. They can determine the outcome of an action without assigning any motivation to the DM and without playing favorites. The extent to which you use them is entirely up to you.
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Some DMs rely on die rolls for almost everything. When a character attempts a task, the DM calls for a check and picks a DC. As a DM using this style, you can’t rely on the characters succeeding or failing on any one check to move the action in a specific direction. You must be ready to improvise and react to a changing situation.
Relying on dice also gives the players the sense that anything is possible...A drawback of this approach is that roleplaying can diminish if players feel that their die rolls, rather than their decisions and characterizations, always determine success.
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One approach is to use dice as rarely as possible. Some DMs use them only during combat, and determine success or failure as they like in other situations...With this approach, the DM decides whether an action or a plan succeeds or fails based on how well the players make their case, how thorough or creative they are, or other factors.
A downside is that no DM is completely neutral. A DM might come to favor certain players or approaches, or even work against good ideas if they send the game in a direction he or she doesn’t like. This approach can also slow the game if the DM focuses on one “correct” action that the characters must describe to overcome an obstacle.
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Many DMs find that using a combination of the two approaches works best. By balancing the use of dice against deciding on success, you can encourage your players to strike a balance between relying on their bonuses and abilities and paying attention to the game and immersing themselves in its world. Remember that dice don’t run your game — you do.
Heavy stuff...but it shows that there different styles of how to handle dice as part of the game. In High School, I would play with a couple people as we walked between classes. No dice what so ever, just describe your action and the DM would let us know if it worked or not. It was combat light, with more puzzle and exploration, but it was fun.
But if the DM ever said, "Okay, roll a d20 to see how well you did," we would literally have to stop in our tracks. Who has a d20 handy with an arm full of books? This sudden change of mechanics would make us wonder what else would be sprung on us. Sort of like the pop quiz we were about to walk into...
Back to Session Zero
You mention, "As a DM I sometimes skip certain resolves..." That would normally be an indicator that you've already set a precedence that you don't rely solely on dice rolls. But it sounds like, at least to the player that is complaining, that you haven't made this well-known to your players, or that you didn't foreshadow this situation well enough.
Players need to be aware not just of the rules, but how you implement the rules so no one is caught off guard. Just like in High School, we knew that dice were not going to be used. But maybe the DM could have added a caveat about combat; "For combat, we need to use dice, so if there is an encounter, we need to wait until lunch so we have a table to roll on."
You need to make clear the rules of engagement; "There are going to be times where I'm not going to let the dice decide. Like if the there is a situation that you can't fail, or if you role play/describe exactly the action needed to succeed. There may also be time where you'll need to succeed on a lot of checks to create a winning situation so that the end is worth all the trials."
Now with that ground work, you can play on.
Then during the game, give hints that what they are doing now will make things easier later; "Since you did x, I'll say that y happens." This could be disabling the alarm system means that if the party encounters guards, they can't easily call for back up. Or that they cast a silence spell in a clever way, no one will accidentally walk into it and get suspicious.
This also goes for dice rolls. "Since you rolled so high..." as the hip DM's say.
There is always one in the crowd
In the comments on Jorn's answer, you call out the one player:
- "picking the best possible multi-class combination and min-maxing everything"
- "maybe he doesn't like an enemy getting killed without resolving, because in such situation his rule-exploiting damage chains don't get to shine"
- "I'm not judging a play-style"
One, I think you are judging, but that's besides the point. Two, you're not going to please everyone. But you can at least min/max the damage (see what I did there) by making sure everyone is on the same page as far as how you do your game. If their character only "works" because of dice rolls, and that's how they want to play, then letting them know ahead of time that there are situations where you don't care about dice, you care about role play, then that player can make choices.
I have characters that work very well due to how classes/feats/spells/features/items work in synergy. But that character also has a complete back story explaining WHY they have that combination, so even if I don't get to roll 20d6 damage because the rogue got in a coup de grace, I still have a character I can speak through.