Thumb-operated levers on trumpets are uncommon (though, IMHO ergonomically superior). More common are a ring in which you place your left ring finger. The ring is directly attached to a slide on the third valve, so you can flatten notes by extending your left ring finger.

What I've seen is a thumb lever for the 1st valve, finger ring for the 3rd valve.

Those removable slides are usually just to get the spit out of the loops. I haven't seen the lever you mention - are you sure you are not just looking at the usual [water key](https://www.youtube.com/watch?v=vMbb8-WK_VM)?

With valved brass instruments you are trying to approximate a logarithmic relationship with a linear sum of components. Trumpets have a high resonant Q, so not using the valve slides is going to produce out of tune notes. I played horn once upon a time. Horns have low resonant Q, so you just “lip it in”.

one of my favorite aspects of learning the tuba was when we covered logarithmic approximations via linear summing.