Most quant heavy/technical desks

This is very misleading and not what xVA desk does - there is no prop trading and there is nothing involving convexity adjustments. Here's a counterexample: bonds have convexity and yet there is no xVA involved for this product because it is not a derivative. Convexity adjustments refer to something completely different and most applied to interest rate swaps with mismatched settlement and reset dates (vanilla interest rate swap vs. constant maturity swaps).

Let me clarify the definition: xVA are a series of valuation adjustments involving any type of derivative portfolio to account for the credit risk (formally known as counterparty credit risk) of the issuer engaging in a long derivative position from the broker-dealer (BD) [known as CVA - credit valuation adjustment], the BD's own credit risk (DVA - debt valuation adjustment), and a series of other adjustments related to margining (MVA), funding (FVA), capital (KVA) and so on.

In simple terms, the BD who is short a derivative contract collects a premium (this part you mention is accurate but not for the reasoning you suggested) because of the riskiness of the issuer might default on its contractual obligations related to the derivative (think final payment of a forward, margin requirements for futures, quarterly swap payments and so on) known as CVA. In return, the BD will also pay a premium to account for its own default risk on the same types of payments called DVA. Typically DVA CVA and these adjustments are netted.

Complicated quantitative credit models that take into account loss given default (LGD), probability of defaults (PDs), expected exposure (EE), discount rates, and so on are all used to price CVAs/DVAs and similar computations for other adjustments. Simple models will only have constant risk neutral probability of defaults or constant yield curve assumptions but the more complex ones involve simulations run in code to predict future exposures and various default scenarios using stochastic default probabilities and dynamic yield curve (short rate) models.

Now that we have a clear understanding of xVA what do these desks actually do? Traders look at the outstanding derivative contracts on the BDs book and hedge out these counterparty credit risk exposures through credit derivatives (CDS, CDX) and might invest in the excess cashed received from CVA in relatively safe money market securities. The quants on the desk will support the traders in pricing these xVAs, answer technical questions, and build analytics such as simulation models in C++ for pricing and exposure prediction.

It is a very quantitative desk and a thorough education (masters in financial engineering or STEM PhD for the quants/traders, or really strong undergrad in STEM from a top school for a trader) and understanding of derivatives across asset classes with the corresponding risk exposures are a must. This desk is not the sexiest, and much different from your typical derivatives desk where understanding flows, market moving events, and strategies is necessary. Rather, it is closer to a risk function for the bank given it is not market making the product but instead understanding the credit risk of the BDs derivative portfolio and hedging out this exposure (purely a hedging desk, no speculating).