When someone writes @yourbrand, the platform notifies your account and the social team sees it. Those posts are addressed to you, so they are polite more often than not, and most of them are support requests. They are worth answering, but they are not where reputation moves.
Untagged posts are different. A customer complaining to their own followers, a reviewer comparing you with two competitors, an analyst quoting your pricing in a thread: none of these reach your notifications. They are written for an audience, not for you, which makes them more honest and more likely to be reposted. A Twitter monitoring tool exists mainly to find these.
MentionRadar does not need access to your X account. It looks for public posts that contain your brand terms as whole words, whether or not the author tagged you, and shows tagged and untagged posts in the same feed. The two posts on the right are from the same day about Kestrel Bikes. Only one of them would ever have reached the brand account.