If you're using the rules in XGTE, tool proficiency when you're already proficient grants advantage.

Realistically, Dragonchess, as described (and published) is a complete information game, just a quite complex one, which drastically limits the value of deception -- at higher skill levels it doesn't really matter what the other player wants to do with a given action, all that matters is what the action permits. If a given move is threatening, you have to react to the threat even if the other player didn't intend or even notice the threat you see. Your rules are better for incomplete information games (commonly card games, though there are a few incomplete information board games, such as stratego).

The issue is that no specific skill in D&D was designed to represent games without luck. Intelligence is an ability score and it works, but it's kinda boring to play a game as rolling one of your 6 ability scores instead of doing a skill challenge. Additionally, it's way harder to make it complexer, more interesting, engaging, and less simplistic.

But the issue is that none of the skills really fit, at least in my eyes. So yeah, deception doesn't really fit. But what does outside of intelligence and previous knowledge of how to dominate Dragonchess.

I love this idea and have an amended version of this that I think works well and covers everything. 

Key skills - Investigation, Insight, Deception.

Investigation beats Insight.

Deception beats Investigation.

Insight beats Deception.

If you win the rock paper scissors you roll with advantage. 

All players roll an Intelligence check and add their proficiency bonus if they are proficient in that skill. So it would be an Intelligence (Deception) or an Intelligence (Insight) check. 

If a player has dragonchess proficiency they add their proficiency bonus to all options. 

So this could look something like :

D20 result + intelligence modifier + proficiency bonus (insight) + proficiency bonus (dragonchess)

Or advantage D20 result + intelligence modifier + 0 (no applicable proficiencies) 

For more advanced mechanics and a bit of my on flavour read on! 

I was also thinking there could be an element of cheating baked into the rules involving Sleight of Hand. You can opt to contest Dex (Sleight of Hand) Vs Wisdom (Perception) (I'd rule advantage to Perception). On a success you gain your proficiency bonus to your final roll. On a failure you take a -5 penalty to your final roll.

In this version of the game cheating is a valid strategy in dragonchess which can give yourself a positional advantage but comes at a cost. If you are discovered cheating, the opponent gets to remove a mid value piece (or lower) of their choice for free.  More expensive dragonchess sets are enchanted to verify whether a piece has recently been illegally moved and by which player. A rare spell called Emalkhaxhazh's Impartial Adjudication (4th Level Wizard spell) is required to enchant a dragonchess set in this way. 

This is honestly Awesome, Thanks for the time and Effort you have put into it.

In the 2024 rules Dragonchess is tied to Wisdom, but I liked OP's three contested skill checks as they're pretty similar to the rules of gambling in XGE (Page 130). I just removed the "If player A uses Deception and player B uses Insight" math to make the rules lightning quick that doesn't really require thinking. Naturally, these rules don't really translate to how Dragonchess is actually played, but I wanted a version for quick resolution. Here are some variations I've come up and played with:

Dragonchess (Requires Proficiency)

Version 1. (Quickest, basically just brute forcing skill check against skill check)
First player to win three rolls wins the match
Players roll a d20 and choose any of the following
Play – Roll a Wisdom (Dragonchess) check
Bluff – Roll a Charisma (Deception) check
Read – Roll a Wisdom (Insight) check
Reveal rolls, highest total wins the round, reroll ties, start from the beginning until a victor emerges
(Optional: Limit the consecutive use of the same roll so that Bards and other high charisma classes can't just blast through matches by spamming Deception)

Version 2.
First player to win three rounds wins the match
At the start of each round, both players secretly arrange the three approaches in any order
Play – (Dragonchess)
Bluff – (Deception)
Read – (Insight)
Both players reveal their order and start rolling and revealing the contested checks, if at any of the three steps, one player’s choice is Deception and the other’s is Insight, the Insight roller gains Advantage for that roll
Count the total of all three rolls together, highest total wins the round, reroll ties

Version 3. (Not that simple anymore but a bit spicier)
First player to win three rounds wins the match
In each round players choose one of the following and rolls:
Play – (Dragonchess)
Bluff – (Deception)
Read – (Insight)
Each player has two Dragon tokens per match they may spend (max one token per round) to do one of the following:
Reroll your die
Add 1d4 to your roll
(Optional: Combine and add the optional rules listed above. Limit the consecutive use of the same roll and/or if one player rolled Deception against Insight, have the Insight roller gain Advantage)