It's a very thin line to walk, and I don't think "true neutral" or "true unbiased" exists for the same reasons you mention. A newspaper may be neutral and stick to the facts, but if it simply chooses not to publish something, or it chooses to publish something as a byline instead of headline, it's already biased.

There are organizations trying something else though; admittedly I got this through a promotion from a youtube channel, but Ground News [0] is a news aggregator that publishes news with many sources, including ownership, political affiliation, publishing type (facts or opinions, neutral or entertaining), and even notes on which channels do and don't report on it, identifying media blind spots like "low coverage from right sources" or "low original coverage" to help people escape their bubbles and provide multiple perspectives on an issue.

[0] https://ground.news/