Honestly, the answer is no. Not for any notable period of time and reasoning is below

Seeing major improvements in PS5's situation does not mean it besting Nintendo in any category. Historically PS big hitters are third party... There are hardly any third party games with amazing legs, even on the Switch. We have Minecraft, Momotaro, Monster Hunter, a few indies like Humans fall flat. The charts will be continue to be under Nintendo/Pokemon's spell for the forseeable future but that doesn't equate to Playstation doomed.

Lets look at Playstation 2's top selling games, baring in mind it sold 20m versus PS4's 9.5m & that the PS4 has digital sales which are not accounted for. We all consider the PS2 to be a huge success in Japan but its software doesn't come close to the Switch:

1. 3,555,000   PS2  Dragon Quest VIII: Journey of the Cursed King

2. 2,325,000   PS2  Final Fantasy X

3. 2,323,000   PS2  Final Fantasy XII

4. 1,961,000   PS2  Final Fantasy X-2

5. 1,615,000   PS2  Dragon Quest V: Hand of the Heavenly Bride

6. 1,439,000   PS2  Gran Turismo 3: A-Spec 

7. 1,197,000   PS2  Dynasty Warriors 4

8. 1,129,000   PS2  Kingdom Hearts II

9. 1,117,000   PS2  World Soccer Winning Eleven 6: International

10. 1,100,000   PS2  World Soccer Winning Eleven 7

Playstation has never had an abundance of mega sellers, that is simply not a metric to meassure its success against. PS4's userbase is about half of what PS2's was and that is more or less reflected in its top sellers. The point is that the expectation for Playstation hardware (around 10m) does not require games that compete with what Nintendo is doing. Nintendo's output is just unmatched and Sony has never come close.

In a scenario in which the Playstation 5 more or less stays equal with PS4, you'd see  2 or 3 1m+ sellers and the an abundance of games around the 500-800k mark. That doesn't spell trouble, it just reflects that market Playstation has been working with since the PS3's release.

Comparing PS5 to the Vita is also misjudged. The Vita's biggest game was a port of Persona 4. There was nothing on Vita comparable to mainline Final Fantasy, Kingdom Hearts, Monster Hunter, Dragon Quest etc. Playstation will get all of those and much more.  Elden ring was a break outs success and there will likely be more of those titles to come. 

On the topic of PS5 games, very curious to see how Harry Potter and Avatar peform in the Japanese market.

My argument is about competition.  The PS2 wasn't competing with the Switch, so it didn't need games that sold like Switch games.  It was a different era with different sales levels.  The PS2 was competing with Gamecube, XBox, and Dreamcast.  None of those consoles had software sales like the PS2 games you listed.  They couldn't compete.  That is why they sold poorly.  

For the same reason the PS5 will continue to sell poorly.  It can't compete with the Switch.

The problem with PS5 mantra is doesn´t count Sony platform's declines in years, and now is very visible. Sony not losing Japanese market now, but losing over the years.

First losing PSP and PSV pipeline production ( great for their home console), with losing exclusivity of A and AA production Sony losing the handheld market too.

Losing the indies exclusivity too.

Now rest some AAA production isolated. 

Switch has indies, A and AA productions and All Nintendo production in one pipeline vs AAA Japanese production ( and Capcom maybe rethink world production). This a reality of the market, and some AA+ and AAA have launched on platform. 

AAA Japanese is isolated from the Japanese market, not Nintendo. 

Famitsu Sales: Week 32, 2015 (Aug 03 - Aug 09)

Top 30

3DS - 16
PSV - 6
WIU - 4
PS4 - 3
PS3 - 1

Famitsu Sales: Week 27, 2022 (Jun 27 - Jul 03)

Top 30

NSW - 25
PS4 - 4
PS5 - 1

This seems more like a spiral of death textbook than rising in two years, without analysis context, macroeconomics, etc. Macroeconomics looks grimmer from Sony. Only the core market buys one console for waiting for one game, the norms play games with people buying, with games people playing, the dominant and popular console in the market. 

But does the PS5 need to compete at the Switch's level in order to sell 8-10m? I don't think so. It'll have a a strong selection of games that will not come to the Switch as listed. Ultimately they're fulfilling different interests. Will the Switch eat into Playstations audience? Probably, but not by an amount which stops PS5 from succeeding. 

It's obvious at this point that a significant portion of PS5s "sold" in Japan are not actually in Japanese homes, hence the awful software sales.
This is a result of the global supply situation; if PS5 was available in sufficient numbers both in Japan and globally, it wouldn't be getting scalped and exported like this.

I'm not saying PS5 is going to do amazingly in Japan or pose any real opposition to the Switch's dominance, but taking the current numbers as a measurement of how its going to do long term just isn't realistic. The current circumstances are highly unusual and impermanent.
The chip crisis will eventually alleviate, and Japanese third parties are not going to abandon Playstation because the big ones still sell well on it due to overseas sales shoring up declining local sales.

I agree PS5 will keep flopping in Japan

Will get all the games regardless. There is no reason to not port their games to PS5. Bigger userbase = More sales. With Switch being the lower factor in mind upgrading it to every other platform is going to be easy

And the big-budget games will still get released on PS5 because PS5 is smashing outside Japan, so Capcom and Square have nothing to lose by making their AAA games to skip Switch and focus on Playstation/Xbox/PC crowd, for Switch they will release smaller budget games until Nintendo decides to release a more powerful console

This will prevent PS5 from complete irrelevance, there will always be a small market for Playstation (5-6 million at least)

The seems to be a misunderstanding on the Japanese region here AAA titles have not been a key factor since the Wii, only Monster Hunter and DQ are really of any significance among them and those didn't halt the decline of the PS brand and mind you this was with one half of the PS4's life being against dead competition in the WiiU.

Getting AAA titles is not the problem appealing to the region is and so few of the AAA titles do so even FF has dropped off. This is before the issue of the elephant in the room, the Switch's successor, it will more than likely be capable of handling such titles anyway while getting titles that do appeal to the region this is the long term issue being ignored. 

You are asking the wrong question.  The right question is "what will people looking for a home console buy"?  In the previous generation the options were PS4 or Wii U.  Given those options about 9-10m bought a PS4.  Today the options are PS5 or Switch.  Those same 9-10m people are not going to show up for the PS5.  A good chunk of them are going to just have a Switch only and be done.

People didn't skip the Switch just because they apready had a PS4. The narrative that people only buy one console doesn't exist in other markets, not sure why it will exist in Japan when the Switch still fundamentally lacks the majority of PlayStation software. Famitsu's most demanded game is a PS5 exclusive. Everywhere else we see the Switch sell alongside PS5/Series X. Especially with the 4 year gap between them, it doesn't have to be either or... People who bought a switch 4 years ago, can be partial to buying a PS5 today.

We also have to acknowledge that the Switch is appealing to an expanded audience too. I dare say the majority of the audience were never going to buy a playstation in the first place.

No one disagrees with the notion that Switch will eat into playstations audience, the point is that

1. Playstation will still have major support from 3rd parties that the Switch will not.

2. This will secure playstation 5 a decent future which will realise itself once stocks allow. 

Honestly more meaningful then the back and forth would be for people to just give their predictions for Playststion 5 hardware. I say 8-10m, what do you say?