I assume you mean a nosewheel that's not aligned straight ahead by it's steering mechanism, and not a mechanically crooked nosewheel. I'll bite...why not?
Why not use the parking brake?
I don't know that teaching the student to stop and set the parking brake is warranted...especially at a busy field where stopping to execute the checklist is definitely not the norm. However, proceedures and technique may be taught which maintain a safe operation. Verifying out loud the flaps, pointing to them, befor raising the, as part of a challenge and response checklist is one such measure...even if the student is alone in the cockpit on a solo flight, the student should always be taught to execute the checklist and do it out loud as challenge and response. Standardize.
This can be done on the roll as well as stopped.
If a second person is on board, be it pilot, passenger, examiner, inspector, check airman, or the ubiquitous easter bunny, that person may be used to move a control such as flaps, or taxi the airplane while the pilot moves the flaps, or look outside for traffic, or read the checklist, or any number of cockpit/crew resource management techniques. (I have it on good authority that the Easter bunny is a poor control manipulator, and lacks opposable thumbs, though that is second hand information, so take it as you will).
Waiting until clear of the runway may or may not be appropriate; it's probably generally accepted and appropriate in the training environment. In the real world I've been known to retract flaps while landing or coming over the fence for short field proceedures or gusty tight conditions. I don't teach that to an elementary student, though I may teach and demonstrate it to a company pilot who has the experience and judgement to use such a technique. Likewise, when operating a fabric covered airplane on a loose surface, retracting flaps and configuring while on the runway may be a better choice, and using flaps to return weight to wheels on ice, slush, snow, etc, may be a preferable technique...especially in places where stopping is inadvisable (skis, for example).
I do insist that gear and flaps always require two calls, even if they must be from the same person when there's only one person on board. If two are on board and one starts moving gear or flaps without verification and confirmation, somebody is going to get their hand slapped. Regardless of w(h)eather the aircraft is still in motion.
If the brakes are hot, an aborted takeoff has just been performed, or the aircraft is practicing landings, applying the parking brake may not be warranted, or advisable. I had a brake failure a few weeks ago after a normal landing, a few minutes on the ground and a taxi to takeoff. The brakes were applied while sitting loading on the ground, and when I got to the end of the runway, I couldn't turn left. Enough heat transfer had occured to the brake unit from the disc with the park brake applied (hot desert location with strong crosswinds on a downhill runway at high density altitude, on eastern-bloc brakes) that my left brake faced. I could turn right all day, but not left, and brakes were the only steering. I shut it down at the end of the runway, chocked it, and waited a half hour before I had brakes again.
Setting the park brake on a really hot brake can boil the hydraulic fluid in the caliper (or the expander tube for those who remember them), damage seals and packings, and transfer enough heat to the brake assy that it fades or is disabled. It can also lead to cracking of the pads, warping or impressing of the disc, and overal brake wear, damage, or reduced life. After landing or an aborted takeoff, even if little brake has been used, tire and brake temperature is at it's hottest, and this is the worst time to apply the park brake for any extended period.
If you're using the brake while going heads down to run a checklist, so be it. The park brake is there for a reason, but remember that it has it's limitations.
As far as taxiing clear, okay...but stopping isn't always possible or justified. For primary training, it's probably not a bad idea initially, but it's not real world, and it comes back to the addage that you train how you fight. Train the student to operate in the real world and standardize it that way, and the student has a much greater chance of success in daily operation, and of avoiding lapses in checklists and proceedure in the future.
--Incidentally, doing mag checks "on the go" is probably not a good idea. Adding power while holding brakes is bad form and only causes the brakes to absorb energy in the form of heat that they shouldn't have to take. It causes wear, diverts your attention at a time when both the mag check and taxiing should have your full attention, and doesn't allow adequate time to properly examine the engine responses. If you're only listening for a quick drop and no banging, then you're missing a great deal of what's important in that check. Stop, and do it right.
This is especially true of other checks, such as propeller cycling (where is it governing at the bottom end with barometric power applied...have you ever looked?), feather checks, autofeather checks, governing or governor checks, etc. Stop and devote your time and attention to the systems...and yes, that's a good time to use the parking brake. Using them when the aircraft is stopped, rather than moving, is a whole lot easier and lower temperature, and it's really their forte.
If you're flying something carbureted, and doing carb heat checks on the go, then you really should be stopped and concentrating on the check...which should be much more than a quick on and off check for a drop in RPM or change in manifold pressure. You're checking to see if ice has formed, how far the drop is, and to melt any ice that has formed...it should be a good fifteen second or more check, and shouldn't be done on the fly, so to speak.
