Do you have to have thieves’ tools in order to pick a lock?

Does a character have to have thieves’ tools (TT) in order to pick a lock? Not necessarily proficiency in the tool, but at least have the tool itself?

I’m confused about picking locks and whether or not it A) requires thieves tools to even attempt to open a lock (the description of the lock and manacles in the equipment section suggest they are), and B) if you need proficiency in TT to even use them (the “Working Together” section in PHB 175 suggestes you do). However the dexterity ability section on PHB 177 clearly lists “pick a lock” with no other qualifiers.

Clearly you can’t pick a lock with just your fingers, so some type of tool is needed. I figure that improvising tools gives you disadvantage, while having the TT avoids that penalty. This allows my high dex characters to still attempt to open locks at any time (in a jail, stripped of their gear, etc) but obviously having the right equipment is a huge advantage.

So I ruled that picking a lock without TT is a dex roll with disadvantage, with TT (but not proficiency in them) is a normal DEX check, and proficiency lets you add your prof bonus to the roll (as the TT item entry seems to imply).