Trade bans that don't show on steam profile?

that is a different thing, valve has 3 statuses:

1st stage: Valve Trade Lock
- Cant do anything
- Valve is investigating the account.
- Is temporary.
- is invisible.

2nd stage: Trade Ban
- Cant trade etc.
- Duration of between 3 weeks and perm.

3rd stage: Trade Probation
- Can trade (in most cases)
- But have some restrictions in regards to their account. (activating/buying games etc).
- 3 weeks or longer.

They can also lock the account ( made inaccessible by owner) due results of the investigation.

It can be that from Trade Lock it goes to unlocked. As nobody has the exact policy of valve, and they deviate from it often enough, there is no real way to find out. They don't share their info, and nobody has all their info.

All trade ban/locks are related to:
- payment problems to valve for purchased things
- scamming
- hijacked items going through the account.
- being alts of investigated peoples accounts with above.

tl;dr: the player you linked is currently trade locked, as valve is investigating the account. (You get the trade ban warning because since he's under trade lock, he can't trade, and that's the only warning of that type Valve has)