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The PHB and basic rules describe the bard's Jack of All Trades feature as follows:
[...] you can add half your proficiency bonus, rounded down, to any ability check you make that doesn’t already include your proficiency bonus.
trying to open a lock requires proficiency with thieves’ tools
Assuming that Jack of All Trades does not give you proficiency in any skill that you are not proficient in, does it mean a bard actually can NOT pick a lock?
Yes, it means exactly that. The requirement to use Thieves' Tools to pick locks is proficiency in using the tools, which Jack of all trades does not confer. The bonus only confers a numeric bonus to ability rolls that don't include the bard's proficiency bonus, nothing else.
That being said, the rules on whether the Thieves' tools proficiency is actually mandatory to pick locks are a bit conflicting. See Do you have to have thieves' tools in order to pick a lock? - I won't repeat things said there, it's up to you how you decide to interpret them.
However, the question in the title remains unchanged: Does Jack of All Trades make you proficient with thieves’ tools? Whether or not the proficiency is needed to pick a lock, the answer to that is no.
The Jack Of All Trades feature doesn't give you proficiency, it just allows you to add half of your proficiency bonus to various checks. As you quoted, the PHB specifically says
trying to open a lock requires proficiency with thieves’ tools
It does NOT say "trying to open a lock requires you to be able to add some sort of bonus". Jack Of All Trades does NOT grant any proficiencies, so it does not allow you to do anything that requires a proficiency you haven't got through some other means.
Jack of All Trades doesn't make you fully proficient with tools, but it does grant you a "half proficiency". Looking at the official definition:
Jack of All Trades: Starting at 2nd level, you can add half your proficiency bonus, rounded down, to any ability check you make that doesn’t already include your proficiency bonus.
When you use the official D&D Beyond character sheet, all non-proficient skills for a bard with Jack of All Trades are marked as Half Proficiency in the tooltip. Since the rule says that the Jack of All Trades bonus applies to any ability check, it applies to a Tool check since using a tool is also an ability check.
You can see on this answer from Jeremy Crawford on Twitter that it even applies to initiative, since that is a Dexterity check. And in fact, the auto-calculated Initiative bonus on D&D Beyond does include that bonus:
@JeremyECrawford: Yes, Jack of All Trades can apply to initiative, since that roll is a Dexterity check.
In the thread linked above there is an ongoing debate about whether or not you can use Thieves Tools to pick a lock when you are not proficient. First, a half proficiency can be considered a form of proficiency, just weaker. Second, in this other Twitter thread, Mike Mearls confirmed that anyone can use the tools, proficiency just adds the bonus:
@mikemearls: anyone can use tools, prof bonus adds if proficient. No tools, DM can say no check or disadvantage.
In an effort to be thorough, I asked Jeremy Crawford on Twitter and got a confirmation response from him:
@JeremyECrawford: Proficiency isn't required to use a tool, unless that tool's description1 says otherwise. Other things in the game might make exceptions and require proficiency, but those exceptions have no effect on how tools work otherwise. #DnD
In the same Twitter thread, Jeremy Crawford re-clarified the same thing in different words in a separate tweet:
@JeremyECrawford: You don’t need proficiency with thieves’ tools to use them. If you did, the description of the tools would say so. Some locks require you to have proficiency with thieves’ tools to unlock them. That is a property of those locks, not of locks or tools in general. #DnD
To Summarize: Does Jack of All Trades make you proficient with thieves’ tools? Technically, no. But unless a tool says otherwise, anyone can use it and roll an ability check. Since Jack of All Trades applies to ALL ability checks where the proficiency bonus is not already applied, a bard using such a tool does get to apply their "half proficiency" bonus. The only exception is if a specific lock calls for a proficiency requirement, then technically a bard without full proficiency in Thieves' Tools wouldn't be able to pick that lock. It would then be DM's option to decide if "half proficiency" is sufficient.
1The tool's description, which does not say otherwise:
Thieves’ Tools. This set of tools includes a small file, a set of lock picks, a small mirror mounted on a metal handle, a set of narrow-bladed scissors, and a pair of pliers. Proficiency with these tools lets you add your proficiency bonus to any ability checks you make to disarm traps or open locks. (Basic Rules, 2018, p. 53).
