It’s for tuning
Trumpets don’t need it much until you get into the top end of the range, where certain notes are out of pitch. 5th partial is particularly flat, and 7th is sharp. Using 3rd valve as a replacement for 1+2 is flat, so you use the slide to correct.
Doesn’t really matter until you get to a pretty high level, are playing in a large brass band or know your instrument well enough to know which notes tend to be a few hz off.
Weather can also be an exaggerating factor, so playing indoors won’t have the same need
I’d say the use of the 1st and 3rd slides can be taught pretty early on. The low D is sharp and needs a bit of slide, and the low C sharp is even more sharp (can be off by 25 cents or more depending on the horn) and needs more slide. These are on the mid-low range of the instrument and are some of the first notes that beginners learn to play, and they sound pretty wildly sharp without a bit of adjustment. Not that you’d need to insist on it to beginners, but it wouldn’t hurt to at least mention that the real fingering for a low D is 1+3 + slide.
In my experience when folks say their low Ds and C#s aren’t sharp, it’s usually because their main tuning slide is too far out and they’re playing everything way flat, or the horn just has major tuning issues.
Weather can definitely affect tuning, but it affects it pretty much uniformly. If it’s hot out the overall pitch will rise, if it’s cold the pitch will drop. You just adjust your main tuning slide accordingly and continue to use your valve slides the same way.
All this is to say I think the use of the slides should be part of the fingering chart. It doesn’t take much energy to learn and it makes a huge difference.
Source: I never grew out of being a band kid and play trumpet for a living.
