It's very easy to ascribe all these problems to a "general" Japanese incompetence with technology, but I don't think that this simplified explanation stands up to scrutiny.
Japan was a world leader in many fields during the 80s and 90s, and even up until the late 00s, Japan was considered to be a leader in areas like Mobile Internet (this mobile internet was quite crummy in practice, but it was better than what was available on western feature phones at that time). Even today, there are many pieces of Japanese technology are quite impressive.
It's not just toilets, and in fact they are not unique to Japan anymore: you can easily get your hand on equivalent washlets in China and Korea as well, even if they are not as ubiquitous in public spaces. Public transit payment still beats any other large country I've been to: FeliCa-based IC cards are standardized and accepted almost everywhere, they are faster than MiFare-like cards[1] and you can easily recharge them on your phone (the situation on Android is less than ideal, with almost no support on international models and the somewhat typical corporate low-quality botch job that is Osaifu Keitai[2] - but it still works). Vending machines are still world-leading and many of them (though not all) support digital payment through IC cards, iD or QUICPay. Digital payment on highways with ETC is fast and reliable, even if it's not the best system out there.
I feel like Japan has an industry-wide problem with software (especially in terms of development speed, quality and UX), but newer services are coming out with better quality software nowadays. Older internet services which predate the smartphone era, and pre-internet services that are trying to modernize are the ones who are usually having a hard time to adapt.
The government is also very poorly on the digitalization front and this is a well-known issue. In my opinion it's mostly for two reasons: strong consensus-based organizational culture and system fragmentation.
The consensus-based culture is the easy one, and one that most western commentators will rant about. This culture is also the main reason older companies found it hard to update their software offering. It essentially centers around nemawashi[3] - negotiating with all relevant stakeholders that are affected by the change (this could be hundreds of people), gathering feedback, allaying fears and preparing the ground for the actual approval, which becomes a formality.
The requirement for nemawashi is not absolute. When working in Japanese companies, I've often set things going without doing almost any nemawashi, but I am not Japanese, and it wasn't easy. Most Japanese, even those who are strongly enthusiastic about effecting change, would probably find the approach of "better to ask for forgiveness than to ask for permission" very socially awkward to implement. In the end of the day, if you want to change the way that a certain procedure is done in government offices right now, even if the new way is a clear and obvious improvement, you'll have to get very wide buy-in for that. And if you don't prepare that buy-in, a lot of people will instinctively object and try to block or delay it - not because they think your idea is bad, but because they haven't been consulted. If you've been raised in a culture where nemawashi is the norm, not consulting you can be perceived being disrespectful.
In addition to that, why would a mayor of a small city or a department head at a government office want to take responsibility for issues that could happen with your plan, considering you gave no personal assurances to them in advance? The current system might suck, but it works. There may be many issues, but these are longstanding issues that we've been facing last year and the year before that and probably even for the last three or four decades. Nobody would suddenly start complaining about them.
This is exactly what you get with all these "minor integration bumps". These issues are expected to happen in most cases of digitalization. Governments tend to outsource software development and ensuring the quality of outsourced software is not easy when almost nobody in the government has the necessary technical skills to assess the quality of said software. But look the difference between Japan and other countries is the tolerance for these roadbumps. Look at HealthCare.gov. Its launch was an absolute disaster, but the US federal government kept working on it until it became usable. Granted, this disaster should not have happened, and the government should have been better in understanding digital service, but I as far as I know (maybe I'm wrong), there was no pushback for delaying other digital services due to risk.
Besides the consensus-based culture, fragmentation is a big issue in Japan when it comes to government services. A great bulk of the government procedures in Japan fall on municipalities. This includes registration of marriages, births or address change, setting up national health insurance, registering your official seal[4], receiving various welfare benefits, arranging and distributing COVID-19 vaccinations and - yes - handling applications for My Number Cards (which often take quite a while to issue).
During the COVID-19 vaccination effort, this discrepancy was in plain sight, with some municipalities being 2-3 months behind others (and even the fastest municipalities being behind the original schedule). The government organized a centralized system for recording vaccinations, but as far as I understand the municipalities had to handle: collecting the data on the various necessary age brackets (and thus required vaccine allocation to each municipality at each stage), distributing vaccination vouchers, coordinating with local clinics and hospitals to set up vaccination venues, setting up a system that allows residents to book a vaccination time-slot at a venue and finally tracking administered vaccinations. Municipalities vary in size from large cities like Yokohama and Osaka, highly concentrated special wards in Tokyo and small villages with less than 5000 people. Each of these municipalities, regardless of size or situation, had to handle an emergency situation by itself with little time to prepare. As a result, most (if not all) of them choose to arrange everything manually for the first round. Data gathering in preparation was slow and in many cases you could only book a COVID vaccine by a phone call (and of course, the phone lines were constantly busy).
The second round of vaccination (the first booster shot) was a lot more organized. The boosters themselves were delayed, and municipalities vehemently opposed the central government calls to speed up the delivery (citing, again, lack of time), but when the shot was finally ready, it was the difference between night and day: In my ward I could already book a slot at nearby clinic (20 meters from my home) online with little hassle. I could also issue a digital vaccine certificate in Japanese or English from a special app by scanning the My Number card with my phone.
In short, the central government had enough time to improve the centralized vaccine certificate service and the local municipalities had time to set up a vaccine booking and distribution platform (possibly developed by an third party vendor which sold the same platform to multiple municipalities). The main problem is that it took Japan almost a year to reach this stable state. Japanese organization can adapt transformation, but they cannot do it fast and cannot react quickly to change (like the new consensus that the booster should be given after 6 months).
I don't want to be too critical of Japan here. While I think the fragmentation of government service carries very little benefits, it's the product of a decentralized bureaucratic system that functioned quite well in the analog age and now it would be quite a gargantuan effort to centralize everything and train all local government employees throughout the country on the new digital systems. The consensus-based culture is also not always bad. It is bad at adapting to change, but it's usually quite good at organizing complex cross industry initiatives (e.g. IC card interoperability) and dealing with fragmented organizations. It also means that everyone (including citizens, customers and business partners) have less surprises. Even if that includes less good surprises.
If we go back to Tim Cook's offer to have a digital version of My Number Card back in 2022, this seems quite in line with the schedule for similar efforts in Japan. The security concerns may or may not have been valid, but it doesn't really matter. Integrating the My Number Card into Apple Wallet is a pretty big change, and as such it requires the normal gathering of consensus and preparation efforts.
[1] https://atadistance.net/2020/06/13/transit-gate-evolution-wh...
[2] https://play.google.com/store/apps/details?id=com.felicanetw...
[3] https://en.wikipedia.org/wiki/Nemawashi
[4] https://hanko-seal.com/archives/5970