Using typescript-language-server through tramp can't find servers for typescript-mode

Question

I am trying to work remotely on a typescript project, with lsp-mode and tramp, through SSH.

I followed the documentation on lsp-mode web page to set up a remote client. Here is my config:

(use-package lsp-mode
  :commands (lsp lsp-deferred)
  :init
  (setq lsp-keymap-prefix "C-c l")
  :bind (:map lsp-mode-map
         ("TAB" . completion-at-point))
  :config
  (lsp-enable-which-key-integration t))
(require 'lsp-mode)
(setq tramp-remote-path '(tramp-default-remote-path "~/.local/bin/" "/usr/local/bin/"))

(use-package typescript-mode
  :mode "\\.ts\\'"
  :hook (typescript-mode . lsp))
(require 'typescript-mode)

(lsp-register-client
    (make-lsp-client :new-connection (lsp-tramp-connection (list "typescript-language-server" "--stdio"))
                     :major-modes '(typescript-mode)
                     :remote? t
                     :server-id 'ts-ls))

And typescript-language-server is installed on my remote machine, version 0.8.1. I am able to use typescript with lsp-mode locally just fine. For that, I followed lsp documentation for setting up ts-ls. Still, I get the following error when I open a remote project:

LSP :: There are no language servers supporting current mode typescript-mode' registered with lsp-mode'. This issue might be caused by:

1. The language you are trying to use does not have built-in support in lsp-mode'. You must install the required support manually. Examples of this are lsp-java' or `lsp-metals'.
2. The language server that you expect to run is not configured to run for major mode typescript-mode'. You may check that by checking the :major-modes' that are passed to `lsp-register-client'.
3. lsp-mode' doesn't have any integration for the language behind typescript-mode'. Refer to https://emacs-lsp.github.io/lsp-mode/page/languages and https://langserver.org/ .
4. You are over `tramp'. In this case follow https://emacs-lsp.github.io/lsp-mode/page/remote/.
5. You have disabled the lsp-mode' clients for that file. (Check lsp-enabled-clients' and `lsp-disabled-clients').

I think 1 does not apply to me since typescript is supported by lsp-mode. For 2, you can see that I added typescript-mode as major mode in my lsp-register-client. I have checked lsp-enabled-clients for point 5 and it was nil, which meant all were registered. For the point 3, I know lsp-mode has integration for typescript since I locally use it.

Only point remaining for me is 4, I guess. There I suspect my config I shared above might be wrong. Any suggestions?