Unnecessary to trim Adapter?

Dear Galaxy Community,

As we are used to perform quality trim and trim adapter in the very early stage of fastq analysis, isn’t it, in fact, in certain situations not necessary? Such as when the Illumina sequenced read(s) is/are surely shorter than the DNA products (amplicons).

I am aware that the fastq read1/2 files from Illumina sequencer (in my case, NexSeq 150 paired-end reads), reports the read only from the first base pair after the adapter from the 5’ end. So, as my DNA products are all between 200-300 bp long, my reads should not have adapters. Is that correct?
Interestingly, when I performed Trim Galore!, which I intended to remove the target primers and do quality trim), I set-up the adapter sequence as “auto-detection.” There were still Illumina adapter (in my case, Illumina small RNA adapter) being detected. It was even being detected in 40-50% of reads.

Can anyone explain why?

Thanks a lot,