How many ways to pair 6 chess players over 3 boards, disregarding seating arrangement. The problem is how many chess pairs can I make from $6$ players, if it doesn't matter who gets white/black pieces, and it doesn't matter on which of the $3$ boards a pair is seated. 

I have a possible solution which doesn't seem rigorous. Can someone tell me if 1) it's correct (the answer and logic) , 2) what is a different way to reason about it ? Seems very laborious to think of it the way I got there.

I started thinking about all the possible arrangements from $6$ people, and that's $6!=720$. 

Now, if I think of the arrangement $A-B ; C-D ; E-F$, it's clear that within the $720$ total arrangements, I will have counted that same arrangement of pairs with each 'pair' seated on different boards  $C-D ; A-B ; E-F$ etc.. for a total of $3!=6$ per arrangement. So if I divide by that, I will basically take each arrangement such as $A-B ; C-D ; E-F$ and count it only $1$ time instead of $6$  which is what I want $\to 720/6 = 120$.

So far I can think of the $120$ arrangements left as unique, fixed-position pairs. Meaning that for the arrangement of pairs $A-B ; C-D ; E-F$, I know I won't find the same pairs in different order. 

I finally need to remove those arrangements where we have pairs swapped, since I don't care about who is playing white/black. I am still counting 
$A-B ; C-D ; E-F$ and $B-A ; C-D ; E-F$, $A-B ; D-C ; E-F$ etc.. Because each of those pairs can be in one of two states,  $2 \cdot 2 \cdot 2 = 8$ gives me all possible arrangements where each pair swaps or doesn't. So if I divide by that number $120/8=15$ I should get the correct number.
Your solution is correct.

Here is an alternative approach:  

Line up the six players in some order, say alphabetically.  The first person in line can choose a partner in $5$ ways.  Remove those two players from the line.  That leaves four players.  The first person left in the line can choose a partner in $3$ ways.  Remove those two players from the line.  The remaining two players must play each other.  Hence, there are 
$$5!! = 5 \cdot 3 \cdot 1 = 15$$
ways to form three pairs of chess partners.  The expression $5!!$ is read $5$ double factorial.
Yes, you are absolutely right. 

The other approach might be selecting $2$ players out of $6$ since arrangement doesn't matter ($_6C_2=15$).