There is no "right" way to play D&D
Okay, let's go back to the basics.
The play of the Dungeons & Dragons game unfolds according to this basic pattern:
- The DM describes the environment.
- The players describe what they want to do.
- The DM narrates the results of the adventurers’ actions.
The DM might (but don't have to) ask you for an ability check between steps 2 and 3. Players do not declare checks, it's the DM's call. You describe what your character does or says. A DM might ask you for a check before narrating the outcome.
A game might not involve social checks at all! See the DMG page 236 "The Role of Dice":
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.
Another possibility is asking for a roll for anything:
Some DMs rely on die rolls for almost everything. When a character attempts a task, the DM calls for a check a 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.
Both approaches are equally valid. It's up to the DM what playstyle do they choose. Making arguments and appealing to the rules does not help here. If this particular DM's playstyle does not suit you, give a feedback after the game. Say you were not having fun, explain why. A good DM tends to listen to players.
Your particular DM is kind of radical tho
In the worst case, he might be a competitive control freak, who is trying to get a win against players. I hope he is not, however. It is possible that your DM simply has a different background, which is not very compatible with the 5e mentality. Let's see what was wrong with his approach.
In 5e, characters do not "use skills". There is a variant rule in DMG that does not need skills at all. Instead, the proficiency bonus is determined from the character's background. That's correct, you don't "use" your Persuasion. There is no term "skill check" in 5e at all. There is Charisma (Persuasion) check — it is an ability check that adds your proficiency bonus if you are proficient in Persuasion. It is an optional check, the DM might ask you for it when you "try to influence others".
Now, let's analyze the DM's argument:
"if a character cannot be persuaded/deceived into doing something the player doesn't want them to do, deception and persuasion have no use"
That's simply not true.
Rules as written, deception doesn't allow you to force people into believing any bullsh*t you say. It allows you to hide your true motive from the listener:
Your Charisma (Deception) check determines whether you can convincingly hide the truth
Chapter 7: Using Ability Scores
So, when you lose the insight/deception contest, your insight fails, basically. You do not automatically believe anything that was said. Even magical spells like Suggestion do not have such power — they require the words to sound "reasonable".
Persuasion is not magic as well. It can't magically change creature's beliefs, regardless of the roll (if there are opposed persuasion rolls, which I am not aware of). Interpreting it in such ways will significantly decrease the value of mind-affecting spells.
Players do not require a DM in order to to talk to each other
When you are talking to each other in characters, I suggest you not to use dice at all. Instead, just talk! When one PC is talking to another PC, and there're no NPCs around, DM is not involved at all. Therefore, he (she) does not need to ask for ability checks.
Asking for checks and saying to players, what do they feel or think, is a bad practice that can easily spoil the fun. Players can handle this by themselves. Also, it is a great opportunity for actual role playing, which should not be missed.