That’s why we still study all the different clefs. You can transpose by pretending the music is written in a different clef, adjusts the flats and sharps accordingly, and you’re off. The note on the bottom line of the staff is a ‘G’ in bass clef, an ‘F’ in alto clef, a ‘D’ in tenor clef and an ‘E’ in treble clef. So, to play something that is written in G major down a half step, pretend there are 6 flats instead of 1 sharp. To play down a whole step, pretend the treble clef is a tenor clef and the bass clef is an alto clef, pretend there’s 1 flat instead of 1 sharp and off you go - you’re sight transposing. This may seem complex, but it’s way easier that adjusting by interval. Sight reading at tempo means the music is going past too fast for you to really be thinking about the notes, just like when you read this passage out loud, you’re not thinking about the letters, you’re thinking about words, phrases and sentences.
There are two groups of musicians who have the most impressive sense of sight transposition - accompanists who do a lot of recital repertoire, particularly but not exclusively vocal accompanists (‘Collaborative Pianist’ is the modern, politically correct term, but it grates on my nerves.) and French Horn players. For horn players, it is because of the fact that their parts were written for ‘horn in x’, and the assumption was that they would play a horn in that key, or use a ‘crook’ (someone upthread mentioned a slide to change the key of a trumpet - that’s usually called a ‘crook’) to change the key. When the modern valved instrument started to become standard, Horn players had to be able to transpose at sight from the written part to the key of the piece on a horn in F, which reads two different clefs. A horn player I lived with talked about entire lessons spent sight-transposing Wagner horn parts (and Wagner’s horn writing is nuts - he wrote for the old-fashioned instruments with crooks, but he wrote horn parts that would be all but impossible to play on the older instruments. Wagner horn players are kinda like the people who do simultaneous translation.)
The pianists I know who are good at sight transposition all do it by clef/key signature. It’s an impressive skill, but at the same time, it’s part of the job description for certain musicians.
By the way, acsenray, I’m sorry to tell you you’ve got it the wrong way around. When a trumpet player plays a ‘C’ on a Bb instrument, he’s playing a concert ‘Bb’. When you tell a trumpet player to play a ‘Bb’ on his Bb instrument, it will produce a concert ‘Ab’. The example you cite would work for a trumpet in D - that instrument would indeed produce a concert ‘C’ when asked to play ‘Bb’, but you need to write a ‘D’ for a Bb instrument to play a ‘C’.
I tell my students all the time - musicians are nuts! And if they didn’t start out nuts, music drives them that way.