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The trumpet buttons 1,2 and 3 lowers the pitch by 2,1 and 3 halfsteps. So pressing button 1 and 2 gives the same result as only pressing the third button.

So why is the third button so rarely used alone? Why press two buttons instead if you don't have to? I have seen it be used for fast drills that would be almost impossible otherwise, but I cannot understand why it isn't the default for playing the notes E and A.

Button=Valve (wind instrument)
Component of brass instruments

Valves are used on brass instruments to change the tube length to allow a chromatic play. When the valve is actuated, the air is passed through an additional pipe loop of a certain length, extending (usually) the air column of the instrument.

The first valve lowers the natural tone by two semitones (12.4%),
The second valve lowers the natural tone by one semitone (6.0%),
The third valve lowers the natural tone by three semitones (19.1%),

Intonation problems:

If two or more valves are used in combination, the intonation of the sound heard does not correspond to the calculated sum of its intervals, but to a tone that is too high. This is because the additional tube length switched by each valve into the resonator body is only calculated to decrease the air column of the open instrument by a certain interval, but does not take into account the air column already extended by another valve.

The intonation when using multiple valves at the same time is therefore basically more or less unclean if no compensation takes place in any form. Quintessence from this realization: The fewer valves are needed, the better the tone is right.

Possible solutions:

There are basically as few valves as possible to use.
With the small differences of the "short" valve combination (1 + 2), the compensation is usually done by ear via the approach.

As you’re supposing:
The use of the clean handle 3 would be desirable, but in practice usually not possible, since in the fingering tables of the instrumental schools and thus in the training, the combination 1 + 2 in the foreground.

For instruments of rather short overall length like the trumpet, the difference in combinations with the third or first valve is usually compensated by simply pulling out the valve train of the third valve.

Therefore cornets, wing horns as well as high-quality tubes and euphoniums have often already standard triggers as an additional intonation aid:

A trigger means a compensation by a device that makes it easier to vary the length of a valve train. This device, also called intonation pusher or equalizer, consists either of a ring connected to the tuning cable or a U-shaped "saddle" which is moved with a finger or of a mechanical lever which is pressed with a finger and usually a return spring has.

Because valve #3 is normally tuned slightly flatter than #1 + #2 (and will often have a finger mechanism allowing it to be even flatter). This is to do with the physical fact that each semitone down requires the tube length to increase by the same PROPORTION, not by the same fixed length.

The real situation is complicated by most instruments having finger-operated slides on valves #1 and #3, allowing them to be lengthened. But #3 is still normally adjusted to be longer than the sum of #1 and #2, so it would make notes normally played by #1 + #2 too flat.

On a trumpet without 'triggers', low C# and F#, the only notes requiring #1 + #2 + #3, are, if not un-usable, certainly difficult to play in tune.