While there is certainly some validity to this, the roots of this widespread perception are mostly misinformed, as it wrongly attributes the issues at hand. In truth, it relates more to the intregity of the individual's auditory system, the technology level purchased, and the stage of life when the person first started using hearing aids.
Firstly, hearing aids do NOT amplify all sounds indiscriminately; instead, they enhance the necessary sounds through non-linear amplification. Thanks to directional microphones, noise reduction technology, beamforming, and target speech recognition, they constantly scan the sound scape to prioritise speech over background noise.
One common reason individuals express frustration with hearing aids amplifying background noise is related to a condition known as hyperacusis along with people never overcoming the acclimisation period. Hyperacusis occurs when a person's tolerance to sound is diminished, causing their remaining healthy hair cells to become overstimulated. Not overcoming acclimisation is when the person simply hasn't worn their hearing for the correct length of time, resulting in on-going poor tolerance to noise sounds. It's easy for someone who has been largely deaf for 15 years to forget that normal-hearing people have to put up with annoying background noises too!
Your audiologist can always help to reduce unwanted noise if it disturbs you. But remember, a reduction in noise usually means a reduction in speech intelligibility, so keep this in mind. In simple terms, with appropriate hearing aid use and addressing hearing loss proactively before one becomes significantly older, most people can overcome this problem.