I run a competitor to Azuki, Comikey, and I'm happy to offer some insight into this growing industry.

1. Absolutely, Japanese publishers are interested, but maybe not for the reasons you think. They have realized that the Korean (webtoons) and Chinese (webcomics) markets are expanding rapidly into the international market, and
webtoon services are crushing manga services. Even a lot of the manga audience itself is moving towards reading webtoons.
The most scary example for the Japanese publishers is probably Piccoma, a Japan-only (for now) manga/webtoon service launched by Korean webtoon giant Kakao. They entered the Japanese market fairly recently and now are now top on the Japanese app stores, and offer webtoon content as well as all manga. The Japanese publishers have realized, perhaps a bit too late, that they need to spread their content quickly, and that there is a lot of potential in doing so.
I can't speak too much about what's happening behind the scenes, but Japanese publishers have woken up and most of them are no longer sleeping under a rock. They're even making webtoons themselves now... But licensing is still very difficult for Japanese content compared to e.g. Chinese content, and it takes a long time for deals to go through. I hope that answers your question.

2. Comikey is working very hard towards that goal, as I am sure Azuki is too. It's been our goal since day one to become a replacement for aggregator sites, but that is extremely challenging. Some of the biggest titles are made exclusively available to Japanese subsidiaries of the major publishers, other publishers are convinced that making their own app with only their own content is the way to go. Our only way to convince them otherwise is to show them the appeal of a platform like ours (especially in terms of sales), but even then, some content is simply never going to be available to a company like Comikey or Azuki unless you become the size of e.g. Amazon. Note that the bulk of Azuki's content comes from Kodansha's USA subsidiary, which happily licenses its content to many platforms out there. They are an exception to the rule.

3. You are correct that webtoons (Korean manwha or Chinese manhua) were born in the digital generation, and there are huge advantages they have over the Japanese manga culture/industry and its focus on physical releases. Take for example a fun insider fact - many of the original files for manga are basically held hostage by the printing companies, and the printing company must be paid money in order for us to receive a copy of the files for localization. File organization in general appears to be very poor, sometimes files are simply "lost".
Something important to realize though is that companies like Azuki and Comikey cannot easily license most good/popular webtoons. There is a huge webtoon duopoly in Korea, most content is held up (exclusively) by either Naver or Kakao, and they sublicense it to English platforms they have a stake in.
As you mentioned, Japanese manga fans are also shifting towards reading webtoons, even some of our own co-founders admit to doing so ;). Go to an anime convention and look at a few home screens, you'll see the average person has webtoon apps installed, not manga apps.

Overall, it's been intense so far, and the industry is evolving at a scary (for us) rate. It will be interesting to see where we are a few years from now. Personally, I hope manga survives