Absolutely Not

As others have stated, no you can not use a focus (of any kind) while Wild Shaped, even if the animal you morph into has hands, and they are empty and holding the focus. All of the information you posted in the original question provides the answers for you so long as you cross reference them. But a lot of people have missed a crucial part of why it can't be done.

The description of the Druidic Focus says (5e PHB, p. 151):

A druidic focus might be [...] a totem object incorporating feathers, fur, bones, and teeth from sacred animals. A druid can use such an object as a spellcasting focus.

This essentially states that you get to create a focus out of whatever you deem worthy to call a focus; pretty neat.

The description of Material (M) components says:

Casting some spells requires particular objects, specified in parentheses in the component entry. A character can use [...] a spellcasting focus in place of the components specified for a spell [but not costly or consumed components]

This is the start of the unravel. Here it says you can use a focus instead of a material component (so long as it has no cost/isn't consumed). So essentially focuses are just fluff/filler for material components that most people over look anyway, but if you don't overlook that stuff it's still a pretty minimal net gain.

Part of the description of the druid's Wild Shape feature says (5e PHB, p. 67):

• You choose whether your equipment falls to the ground in your space, merges into your new form, or is worn by it. [...]

This portion infers that you get to wear or hold your focus if you so choose when you Wild Shape, which is pretty cool.

The druid's Beast Spells feature says (5e PHB, p. 67):

Beginning at 18th level, you can cast many of your druid spells in any shape you assume using Wild Shape. You can perform the somatic and verbal components of a druid spell while in a beast shape, but you aren't able to provide material components.

This is the crucial spot. It specifically says no material components. If you recall from what exactly a focus is, it is essentially a fancy material component. Meaning it can't be used, not because you can't "hold" it, but because for Game Design purposes they disallowed it.

This is further backed up by the fact that they allow somatic components (hand and body gestures) even if you don't have hands or the appropriate body part to do the somatic component, i.e. a snake making a hand sign. So holding/wearing the material component/focus is irrelevant because it is completely outlawed in RAW.

The druid's Archdruid capstone feature says (emphasis mine):

At 20th level, you can use your Wild Shape an unlimited number of times.

Additionally, you can ignore the verbal and somatic components of your druid spells, as well as any material components that lack a cost and aren't consumed by a spell. You gain this benefit in both your natural shape and your beast shape from Wild Shape.

Doesn't this seem a bit redundant? It is because it is trying to show you that it is essentially an improvement to Beast Spells. At this point, once you reach 20th level you no longer need the material/focus component. So even at level 20 the answer is still technically no, but it becomes irrelevant because materials/focuses are no longer even needed.

However, this is D&D - so if the DM says, "Well that's stupid, you should be able to use your focus as long as you are wearing/holding it"... Then boom, there you go: the true answer is "yes, you can, so long as your DM handwaves it".

The way I've always thought of levelling up is this: It's an an abstraction of what would really be an ongoing process. A character doesn't instantly get better at a bunch of things once they've killed that 37th orc, they've actually been improving continuously the whole time, so leveling up us really just a 'tipping point' where the increase in skill becomes consistently effective.

In the case of a fighter, it would be them properly internalizing in muscle memory a technique they've noticed makes it a little easier to score a hit in some situations. They've known about it for awhile, but they don't always pull it off right (represented by random dice rolls). When they level up and get that extra BAB, they've got it down, and can do it every time.

For a wizard's spells, I imagine a wizard is constantly experimenting and doing research 'in the background'. When they level up, and get a new spell they've actually been working on that spell for a long time. They've only just has that final 'ah-ha!' moment to make the spell effective and reliable - good enough to use consistently. They still can't use it right away, because they need to finalize their notes and prepare it.

I also figure that levelling up doesn't neccessarily mean they can do the new thing 100% perfectly. +1 BAB (usually) doesn't mean you hit some enemies 100% of the time, so dice rolls still represent trying a new thing and failing. Things like feats and class features fit into this less well, but I like to think this explains things well.