Usually you'd vary the concentration of boric acid being injected in the reactor's core instead, since that doesn't involve wear and tear on safety-critical elements of the reactor.

Nuclear reactors absolutely can vary their output to match demand, this is what France has been doing for 50+ years (and what Germany was doing before switching back to coal). It's not as reactive as coal/gas, but you can still vary within 30-100% of output power at a speed of 5% change per minute. Way more than enough to react to 1-day-ahead forecasted supply/demand, and way more than enough to react minute-by-minute if you've got a tiny bit of storage to stabilize the grid's frequency (e.g. pumped hydro).