The base rules only say you don't get an encounter in the street because there are no encounters to get; they shouldn't be taken verbatim to restrict players from entering gates in the street. There are environment/rumor cards that allow or require an action of the investigator during the AE phase while in the street. If you can do those, then you can certainly enter street-gates voluntarily.

You could also take the line that a gate replaces the street as it would a location technically making the street a location with a gate.

Tibs said:

Those cards, however, specifically state that such an action is allowed in contravention of the universal rules. For example, Disturbing the Dead reads: " If a single player discards 2 gate trophies during the Arkham Encounter Phase while in the Rivertown streets, return this card to the box."

Like I said in my original post: It's fairly clear how the moving gates we're supposed to work. It's just that they don't actually work that way when you apply the RAW.

How I'm pretty sure we all think they're supposed to work:

(1) When a Mythos card shows the symbol on the gate, they move 1 space following the arrow matching the color during monster movement. (They do not follow any other rules of monster movement.)

(2) They cannot move into a space which already has a gate. They cannot

(3) If they move into a space with an investigator, the investigator is sucked through the gate as if a gate had opened on their location.

(4) If they move out of a space with an investigator who had an explored marker, that investigator immediately loses their explored marker.

(5) Only gates in unstable locations can be sealed.

(6) Investigators can enter gates in a street location during the Arkham Encounter phase.

Veet said:

While this is generally a useful way to think about gates for people who having people grokking how encounters work, it does have several problems:

(1) The rules never actually say that a gate "replaces" a location.

(2) This is a good thing, because if a gate actually "replaced" the location you'd run into several problems. First, as noted above, you wouldn't be in an Arkham location if the gate replaced the location which means you would never be able to enter any gates according to the RAW. Second, if a gate had "replaced" the Woods, what would you do when a different encounter card sent you to the Woods? Presumably nothing, since the Woods no longer exist.

This is why the rules consistently refer to gates opening in locations, not "replacing" locations.

(I believe the FAQ uses the "replace" wording once to explain how closed locations and gates interact. But given the multitude of problems it would create, the FAQ should obviously be ignored here.)