Autoloading MaTeX` and Rubi` via DeclarePackage[] in the user init.m does not work

Question

Former question: Autoloading and adding code to a package in the user init.m

I am using MaTeX, and I found the updateMaTeX[] function] at the GitHub repo. I tried to add this to my init.m in the MaTeX` context, and wrote

BeginPackage["MaTeX`"];
updateMaTeX[] :=
  Module[{json, download, target},
    Check[
      json = Import["https://api.github.com/repos/szhorvat/MaTeX/releases/latest", "JSON"];
      download = Lookup[First@Lookup[json, "assets"], "browser_download_url"];
      target = FileNameJoin[{CreateDirectory[], "MaTeX.paclet"}];
      If[$Notebooks,
        PrintTemporary@Labeled[ProgressIndicator[Appearance -> "Necklace"], "Downloading...", Right],
        Print["Downloading..."]
      ]; URLSave[download, target],
      Return[$Failed]
    ]; If[FileExistsQ[target], PacletManager`PacletInstall[target], $Failed]
  ]; updateMaTeX::usage = "suggested at the MaTeX GitHub repo."
EndPackage[]; DeclarePackage["MaTeX`", {"MaTeX", "ConfigureMaTeX", "ClearMaTeXCache","updateMaTeX"}];

However this leads to two issues:

1. (Solved) The supposedly local variables json``download``target are touched in the MaTeX` context, and picked up by the auto-completion. It's a simple fix to add Remove[json, download, target] to the code, but I wonder why no $ModuleNumber was appended to the symbols.
2. Autoloading does not work as expected. Type MaTeX in a new notebook and it is still undefined; evaluation causes this new symbol to be created in Global` . Typing MaTeX`MaTeX, it is undefined, but become defined as soon as it is evaluated; Global`MaTeX is simultaneously shadowed. At this point, the $\text{M}\!^\text{A}\!\text{T}\!_{\normalsize{\text{E}}}\!\text{X}$ document is accessible, but the symbol MaTeX(MaTeX`MaTeX) does not have any downvalues, and thus is not working.

Similar issues occur with the Rubi` package. Therefore I wonder:

1. Is using BeginPackage["Package`"] outside the package .m a good practice?
2. Why DeclarePackage does not work as expected?

---

Related Questions

1. Automatically load packages at startup