Why isn't testing fuel flow from both tanks a run-up checklist item?
There is a very good reason not to run this test: If it is not done perfectly your engine can become starved of fuel during takeoff.

For simplicity let's assume your fuel selector only has two positions (Left and Right) and think about what's involved in the test and a few things that can go wrong:

What are we testing?

One of the points of an engine Runup is to verify that the fuel system can deliver a fuel flow sufficient to allow the engine to develop takeoff power. There are other things we're verifying but let's stick with that item for this discussion.

How can we verify this?

We verify that the fuel system has adequate flow by running the engine up to something resembling takeoff power (most runup checklists don't have you go to full throttle, but you advance the throttle enough that the engine is demanding a significant fuel flow compared to idling and taxiing around).
To verify that your fuel system is allowing sufficient fuel flow from each tank you would have to switch tanks during the engine run-up (with the engine at runup power) and wait long enough to exhaust any fuel that might be in the system between the fuel selector and the engine.
If the fuel selector handle comes off in your hand or the engine quits you found the problem on the ground, so you can get a tow back to the ramp and get the fuel system fixed.

What can go wrong?

The test above sounds innocent enough, but it's deceptively dangerous.
Here are two possible failure modes that may make you rethink it:


The second tank doesn't have adequate fuel flow.
This could be a complete obstruction of the tank pickup or just a restriction in one of the lines, but for some reason the second tank will not supply a fuel flow sufficient for takeoff and climb out at full power.
You may miss this if you don't wait long enough at runup power after switching tanks to drain the fuel from the system forward of the fuel selector, and now your engine suddenly got quiet as you're rolling down the runway - or worse it quits 100 feet off the ground and now you're having a Really Bad Day!
You missed the fuel selector detent.
Easy enough to do, especially if fuel selector maintenance has been lacking and the positions are not really distinct (when is the last time you *really, critically, felt the clicks on your plane?). It could happen in the initial switch, or when switching back to the first tank if that's your procedure.
A fuel selector not in a specific tank position may still allow enough fuel flow to complete a runup when combined with the fuel left forward of the fuel selector, but it may be too restricted to allow a takeoff. The fuel selector may also vibrate into an OFF position during takeoff.
In this scenario maybe your engine stops at 200-300 feet, but you're still having a Really Bad Day.


In both of these scenarios the engine is still running because there is fuel in the system between the fuel selector and the engine (the two big reservoirs are the fuel strainer / gascolator &amp; the carburetor float bowl, but there are also fuel lines that will need to drain before the engine dies).
To get an idea how long these reserves last try setting your engine to Runup power and turning off the fuel selector. You may be surprised how long the engine will run before the fuel after the selector is exhausted. Also remember the time will vary based on power setting, mixture setting, and even aircraft attitude sitting on the ground allowing more or less fuel to get to the carburetor from the lines (a nose-high attitude during takeoff may starve your engine more quickly than sitting level on the runup pad).



How can we safely conduct this test?

Making sure your fuel tanks all work is kind of important, especially if you are going to be making a flight near the endurance limit of your fuel.
There is no 100% safe way to conduct this test, but the way I was taught seems to have the fewest possible failure modes:


Start, taxi, Runup, and takeoff should be conducted on one tank.
Typically this is "the fullest tank", but if they're all equally full then either the one the fuel system return flows to if your airplane has such a thing, or just pick one.
By using the one tank for all your ground operations and your Runup you are pretty much guaranteeing that it flows sufficient fuel to allow a takeoff, and that at least this position on the fuel selector works.
After takeoff switch tanks after reaching a safe altitude.
Do this with enough fuel left in the tank you took off on to turn around and get back to where you started in case you discover the tank you switched to doesn't work, and as with any tank switch be prepared for the fuel selector handle to come off in your hand: Know where you'll put the plane if that happens!
If your engine stops a few minutes after you switch tanks…
…you now have a good reason to suspect that the tank you switched to doesn't work.
Per the first rule of troubleshooting, undo the last change you made (switch back to the working tank you took off on) and wait for the engine to get fuel again. When the engine restarts go home: You have a fuel system problem and need to get it fixed.


If after switching back to the working tank your engine doesn't restart…
…make sure you've waited long enough (it doesn't take long, but it FEELS like a long time when the engine is quiet).
If you waited long enough you may have a fuel selector failure. These don't happen often, but they do happen (and this is why you do the test at a safe altitude with a landing spot picked out).
In this unfortunate case run your engine failure checklist and expect to be making that dead-stick landing: Fixing a fuel selector in flight is probably beyond the skills and daring of most pilots.
(Of course you expected this because every time you touch that fuel selector handle you expect it to come off in your hand like I said before, right? Your CFI was like mine and a few minutes after you switched tanks they pulled the throttle on you al the time, so you got this landing made, Right?)



The test process I use and have outlined above is obviously not free of risk, but the risk level is reduced to be the same as any other fuel tank change: You can be reasonably sure that you won't have an engine failure on takeoff when you're close to the ground, trees, people, and other things we try not to hit with our airplanes, and you can pick when you take on the risk of an engine failure to be sure you can safely recover from the situation.