No, passive scores are not floors to active skill checks.
No, specifically to your phrasing, "All active ability checks".
Active abilities, like Athletics and Acrobatics, always carry a chance of failure beyond "nothing happens." A DM will ask for rolls on these checks because their actions cannot be repeated and failure matters.
Passive abilities, like Perception, Insight, and (potentially) History, carry a "non-notice" result. Additionally, these checks are frequently made by the DM when determining what environmental qualities to describe. This is why passive scores are needed. They streamline the game by preventing players from rolling every time a new room is entered or an NPC encountered.
It's relatively well established that passive perception acts as a floor to an active perception check.
This is not the case.
Jeremy's guidance in the link does not suggest that PP is a floor to the skill's entire function. Anything a passive check would notice is not "lost" by a poor active check, as per Jeremy's suggestion, but the two may function differently.
In Jeremy's interview, he is discussing a check with similar DCs or results. This is not always the case.
As demonstrated in LMoP, for example, active and passive skill checks can function differently:
Spotting a secret door from a distance of no more than 10 feet without actively searching for it requires a passive Wisdom (Perception) score of 15 or higher, whereas a character who takes the time to search the wall can find the secret door with a successful DC 10 Wisdom (Perception) check.
To your question:
Are there any rules text that supports this?
The best guidance we know of on this comes from Jeremy's tweets/interviews, which at best offer this as an option to the DM.
Passive skills have different functions than active skills,
Passive Perception is always "on", but does not necessarily produce the same results as an active perception check.
An active use of a skill is a time-consuming, focused application of the skill that should produce different results. Any active roll should always carry a chance of failure.
To follow the PHB (175) example:
A passive check is a special kind of ability check that doesn't involve any die rolls. Such a check can represent the average result for a task done repeatedly, such as searching for secret doors over and over again, or can be used when the DM wants to secretly determine whether the characters succeed at something without rolling dice...
When the players are using a skill in an active manner, they are using it in different ways than the passive score. For example, players enter a room with a secret door and want to use perception.
The DM should provide information about the room, which might include features gated by passive perception scores. They players may decide:
- To find the door right now
- I want to search for the door and will take time
In the latter example, the DM is allowing the players an extended time to attempt an action that carries no catastrophic failure that can be repeated, or is providing information about the room. In DND 3E, this mechanic was called "taking 10." This is a special use of the passive score -- it allows the players to take the average.
In the former example, the players are attempting a single action with an individual result.