To answer your last question first, you are correct, a familiar is not an ongoing magical effect. It is a supernatural being, but it wouldn't show up to a detect magic spell for the same reason a dragon, zombie, or elemental wouldn't. The Sage Advice Compendium question that impinges on this situation is a bit long, so I'll reproduce the text here, and focus directly on familiars at the end.
Q: Is the breath weapon of a dragon magical?
[...] Our game makes a distinction between two types of magic:
- the background magic that is part of the D&D multiverse’s physics and the physiology of many D&D creatures
- the concentrated magical energy that is contained in a magic item or channeled to create a spell or other focused magical effect
In D&D, the first type of magic is part of nature. It is no more dispellable than the wind. A monster like a dragon exists because of that magic-enhanced nature. The second type of magic is what the rules are concerned about. When a rule refers to something being magical, it’s referring to that second type. Determining whether a game feature is magical is straightforward. Ask yourself these questions about the feature:
Is it a magic item?
Is it a spell? Or does it let you create the effects of a spell that’s mentioned in its description?
Is it a spell attack?
Is it fueled by the use of spell slots?
Does its description say it’s magical?
If your answer to any of those questions is yes, the feature is magical.
Let’s look at a white dragon’s Cold Breath and ask ourselves those questions. First, Cold Breath isn’t a magic item. Second, its description mentions no spell. Third, it’s not a spell attack. Fourth, the word “magical” appears nowhere in its description. Our conclusion: Cold Breath is not considered a magical game effect, even though we know that dragons are amazing, supernatural beings.
So we'll ask ourselves those questions about familiars.
While the qualities you listed are supernatural, none of them are magic items or mention specific spells, and none of them are explicitly magical within the description. The touch-spell delivery is specifically using a spell (and acts 'as if the familiar had cast it'). However, in the Monster Manual (page 9) and the corresponding section of the basic rules, telepathy is described as 'a magical ability', and more to the point, the last paragraph of the description of telepathy has this to say:
A creature within the area of an antimagic field or in any other location where magic doesn't function can't send or receive telepathic messages.
So, in an antimagic field, the familiar's ability to channel your touch spells and its telepathic link to you would be disabled, but all its other qualities, while supernatural and amazing, are not magical game effects any more than a dragon's breath is, and would not be disabled. Your DM may decide that the sense-sharing ability is an advanced form of telepathy and would therefore be blocked as well, but it isn't explicitly so as far as the rules as written.
In the case of other effects that block 'magical effects', such as Leomund's tiny hut:
Spell-channeling wouldn't pass through a barrier that blocks magic, as it's actually casting a spell (bullets #2 and potentially #4), and while telepathy is only explicitly blocked by an antimagic field, the description does specifically call it out as a magical ability, so it falls under bullet #5 and would be blocked.
The other familiar abilities would probably not be hampered since they are supernatural but not magical in the aforementioned sense of 'concentrated magical energy that is contained or channeled'.