Yes HTTPS is used. The thread doesn't say that the web app will be completely insecure, instead it says
This effectively reduces the security of your end-to-end encrypted communication to that of your SSL connection to the server
Which means that anyone who can control the SSL connection to the server can now intercept and eavesdrop on your e2ee communications. So who exactly can control the SSL connection?
Well, if a (possibly state-level) attacker controls/compromises a CA, they could issue a fraudulent certificate for the Signal server and attempt to MitM the SSL connection (this threat is limited, but not eliminated, by the use of certificate transparency.) As @multithr3at3d pointed out, TLS inspection proxies at workplaces are a much more likely form of MiTM and could cause problems if your employer was interested in compromising your private conversation. However, in such a case, the employer owns the machine and would probably just install a keylogger on it, so you would have bigger problems.
This threat is amplified by the fact that such malicious modification of the code served can be done in a targeted manner. The server can ensure that only a specific user/client is served the modified malicious code. This significantly reduces the chances of the modifications being detected and exposed.
Actually we want to develop some chat service like signal with web-client, but this article made us confuse about should we ship a web-client or not, can anybody please explain it?
This depends on your threat-model (or rather the threat-model of the intended audience of your chat service). Will those people just be using it for chatting with friends or communicating with colleagues? Or will it be used by whistle blowers trying to coordinate the disclosure of classified information with journalists? You will have to consider whether the risk outweighs the benefits and decide for yourself whether or not to ship a web client.
If its the former, then having a web client will not be a very big issue. This is closer to the use-case of WhatsApp and WhatsApp does have a web client. (Update: WhatsApp has also acknowledged this issue by releasing a Code Verify extension for security conscious users.)
If its the latter, then you had best follow Signal and stick to using desktop clients and apps which can be signed and their integrity verified.
As evident from the comments some people are confused about why these issues don't apply to the app and desktop client.
The reason is simple. The app/desktop client has to be downloaded once. It is also digitally signed so its integrity can easily be verified. The code signing key can be split among multiple developers, potentially located in different jurisdictions, so that one rogue developer cannot release a malicious update by themselves. If you are paranoid, you can also download Signal's publicly available source code and compile it yourself, and then disable automatic updates. If Signal attempts to publish a malicious update, they will have to publish its source code as well, and the chances are much higher that someone will notice the unexplained changes before any damage happens.
With a web app things aren't so simple. First of all, the TLS key is always present on the server, so anybody with access can steal it, and a single government can gain access to it with a subpoena (unlike a split code signing key, which would need the collaboration of multiple governments if the developers are in different countries). Even if the web pages are digitally signed, manually checking their integrity and ensuring they match the publicly available source code every time is absolutely impractical. Secondly, a web app consists of several web pages. Checking the integrity of all of these even once would be a painstakingly difficult task.