Starting from the conclusion:
The subclass is mostly fine from a balance point of view. It lacks proactive play and, for that reason, will probably be considered a good class for players with little or no experience, while it will be left aside for other (more proactive) classes by more experienced players that wish more options and active features.
I will go bit by bit.
It may not look like much against the 5-finger-heart-exploding-punch of the open hand
Being honest, I don't think the Open Hand is overpowered or even the strongest Monk subclass. Drunken Master and Long Death are at least as strong, IMO. Therefore, I think that comparing to Open Hand is a fair comparison and it will be our metric.
3rd Level - Weak
At 3rd level, Open Hand gives the ability to, when using Flurry of Blows, force some negative effect on the enemy (prone, throw away or taking away their reaction).
You can choose your unarmed strikes to deal slashing damage instead of bludgeoning.
While this has some flavor, it's mechanically meaningless in essentially any scenario.
You gain climbing speed equal to your walking speed.
Also not sure how useful this will be. Climbing is essentially difficult terrain as written in 5e, i.e., you would have half of your speed as climbing speed normally. With this feature you can climb twice as fast, which is relevant if you are climbing high places while in combat... Not sure how often this should be relevant. In my adventures I can't recall a single time this would have been helpful.
Also, at 9th level Monks essentially get this feature anyway.
When you are prone, standing up uses only 10 feet of your movement.
Being knocked down is also not that common. I can see this being used more often, but still fail to see the actual relevance of it.
In addition to the bonus granted by your unarmored movement feature, your speed increases by 5 feet. Your speed increases by an additional 10 feet when you reach 6th level (+15 feet), 17th level (+20 feet).
It's rare that any creature will outrun a Monk anyway, thus I don't see this feature as highly relevant. It might help in a few specific scenarios and might be interesting if you are playing, e.g., a halfling which has a base speed of only 25 ft.
So, for a tier 1 adventure, this subclass seems underpowered, particularly because the features seem to be mostly irrelevant or just simply too situational.
6th level - Fair
While extra proficiencies and expertise are hard to come, I'm not sure I want my 6th level feature being just getting an extra proficiency. I would rather have that if I was playing a Rogue or Bard, which are usually the "skill monkeys" of the party.
On the other hand, as I said Open Hand would be a fair comparison, Wholeness of Body is not awesome either. I would call it a draw here. I don't think any of them are a necessary feature or a completely bad feature.
As a side note, having read Cubic's answer, I disagree that, compared to the ability of healing once per day, getting an extra skill proficiency (or, more likely how I would use it - expertise on one skill) is more boring.
11th Level - Fair
Advantage on Dex Saving Throws is usually good. Compared to Tranquility from Open Hand, it certainly seems better, being honest. But tranquility is, IMO, a bad feature, possibly the worst from Open Hand. Dex saving throws are one of the most common and important STs (along Con and Wis), and it's completely free!
Even compared to Drunkard's Luck, this is a fair feature. While the feature from the Drunken 11th level can be used for cancelling any disadvantage (attack, check or saving throw), it can only be used for cancelling disadvantages (not giving advantage) and it requires 2 ki.
On the other hand, I now agree with Cubic's statement: most of your features are now passive. You are giving little day-by-day choices. "When to use this feature?" is a question that none of your features raises.
17th level - Fair (Weaker when compared to Quivering Palm)
Ok, ok. Now I see your point. Quivering Palm is stupidly strong. Period. Yours is similar to Drunken Master's Intoxicated Frenzy, which is fine as well.
From a balance perspective, the subclass is fine
However, it lacks any extra way for the monk to spend their ki points other than the ones they get from the base class. This happens mostly due to the lack of active skills. Monk is already a martial class - which means it gets comparatively less options and activity than spellcasters. There is no inherent problem with that. Champion makes essentially the same for the Fighter class. The immediate consequence is, at least from my experience, that the class will be more chosen and liked by beginner players, and will not be interesting among more experienced players. If that is fine for you, your class should be fine.
Extra justification for the "The subclass is fine from a balance POV"
First, I don't think that a homebrew subclass has to be as good as the best subclass to be playable. It just needs to be... playable.
That said, a point that comes from my experience with the Monk class: the most useful and interesting features are actually from the base class for the Monk. Unarmored defense allows for a quite high AC even at low levels, Martial Arts provides your DPS, Deflect Missile is handy when fighting ranged enemies, Stunning Strike is quite good, Ki-Empowered Strikes allows your punches to hit enemies with non-magic resistances, etc.
Maybe for that reason, my opinion (based on my monk players and the monk I have been recently playing) is that the subclasses are usually underwhelming anyway and "even" the proposed subclass should be fine as it provides some extra utility to a class that already has a good base utility.