What is the role of quants in trade execution in high frequency trading? AFAIR in "normal" trading trade execution is considered a very mundane task. What role can quantitative modelling play in trade execution in HFT?
Re the first part of the question: Quants play no role whatsoever in the actual execution tasks of trading regardless of frequency or whether we talk systematic trading or not. Its done by traders/execution traders (especially on the discretionary side) and not by quants. As your title suggests your focus is on hft, I still would claim quants do not really monitor the execution of hft trades.
However, on the hft side the definitions of quants and traders somewhat merge and the job description becomes somewhat opaque. On the hft side the whole life cycle of a trade on the systematic trading side is highly quantitative in nature and thus a deep knowledge of programming, execution venues, order types, optimizations and a lot more is required of all participating parties.
I assume you mean with quant a person who analyzes, profiles, and tests strategies and works together with programmers to implement trading strategies. Trading strategies can also mean a focus on an execution technique of cash equity on the Nasdaq exchange, for example.
In summary, depending on your definition of "quant", such practitioners are definitely closely involved in the development and implementation of hft execution algorithms, while they probably do not sit there to keep an eye on the actual executions and risk exposures I (it otherwise would be a huge inefficiency and under utilization of core talent). Though most all hft shops fully automatized risk monitoring, I have not come across a single shop who would not have managers and traders constantly monitor exposures and risk limits on top of the systematized risk checks.