Every ASIC designer starts with verification tasks. Verify functionality and learn about testbenches and automation (regression, self-checking). View a lot of RTL files and after a while also RTL design will be one of your skills. RTL and verification (behavioural) are all about a language (HDL) and EDA tools (simulators, compilers, …). But synthesis will link the HDL code to gates. Here digital design insight is needed. If then will be translated in a (priority) mux. A case statement will be an all-equal mux. Comparators, addition, substraction, multipliers, dividers, counters, … . DFT is a gain a step-up. Testmodes are there for functional (silicon) tests, to debug and for test (check if the chip is good after packaging or on the die itself). You need to know how a netlist (RTL translated by synthesis tool) looks like to stitch scan chains together, control all resets and clocks, … . Hence the level of experience and knowledge is much higher than for verification.
Why do ASIC DFT guys get much higher packages than design verification?