The market convention is indeed to price the index of flat term-structure curve equal to the quoted spread for that maturity. This doesn't cause any arbitrage as the Index CDS is fixed coupon and always settled upfront (i.e. the PV of paying the standard coupon and receive expected losses). Settling the index of a full term-structure would make every trade a pain as both counterparties would have to agree on every pricing point on the term-structure.
The quoted index spread is therefore not the actual par spread unless the curve is flat. However it is relatively easy to calculate the par spread by bootstrapping into it, using the index upfront payment as a PV of a fixed rate CDS. Most CDS traders don't have to go to such lengths and are happy looking at the quoted levels in a relatively simple fashion. Correlation desks are more likely to go for a full bootstrapping as the exact term-structure of hazard rate is more meaningful for risk managing exotic products. At least that used to be the case, maybe now it's all about counterparty risk and trying to figure out what is the fundamental value of anything