Maybe this depends on the rifle....
I have 5 rifles and a double barreled shotgun. One rifle is flintlock.
Typically before leaving home I will use Electronic Contact Cleaner with the straw to spray right through the nipples on the caplocks and through the touch-hole on the flintlock. The cleaners is alcohol with a tiny bit of ammonia and the propellant. Then I blow filtered air from the compressor through the nipples or touch hole. Doing this procedure, I will bet money that the gun will go off - except for one. For no good rhyme or reason one of the rifles has had a delayed fire or fail to fire a few times both hunting and first shot at the range. I suspect the breach design is largely responsible. After fiddling around with this rifle for years there is only one way it seems to be ready every time. I snap not one, not two but three caps and then use a small pipe cleaner on the nipple to be sure the hole and channel are clear. Frankly, it is a pain in the donkey to have to do this. However, this is the only way I have had 100% instant ignition all the time. I've tried 10 shot volleys with many other practices and none are totally reliable with the exception of the 3 snapped caps. This is not a fouling shot, no powder is loaded just a snapped cap. Frustrating but this solution as ridiculous as it seems has worked every time for the past several years. Any shortcut and I am taking a chance. So, for reliability you should be very familiar with your own rifle and what it takes to get a sure-fire every time. Do whatever is most reliable.
Next, the fouling shot you asked about. One of my cap lock rifles always shoots 2-inches high and 2-inches left on the first shot from a clean barrel (at 100-yards). All the following shots will be in or around the bullseye in a group. Granted, my "group" runs around 3.5-inches off the bench, but that first shot will be 2X2 outside of the "group". Whether I spit patch between shots or not doesn't effect subsequent shots, only a perfectly clean barrel will put that first one up and over. I could move the sights, but that requires tapping them (non-adjustable). I just don't think I can tap the perfect amount to move it precisely, so I just leave it. The first shot is not going to cause a major miss or non-fatal shot on deer or bigger animals. All the other rifles and the smoothbore will group the clean barrel shot and up to 10 more - the most I have shot without a total cleaning. So again, you need to know your own rifle. Is the fouling shot for dependability or does it have something to do with where the shot lands? If your clean barrel shot hits the bullseye then it becomes your "fouling shot" and you'd need to know where the next few shots will land in case you do need a second or third shot while hunting. We can all say whether we do or don't and why. What really counts for you is if you need to use a fouling shot or not.