It seems that using "matlab:web(-path-to-local-html-file-)" will always open the link in a new tab. According to the documentation (https://www.mathworks.com/help/matlab/ref/web.html), there is no option to force URLs to open in the current tab.
When I checked the MathWorks Documentation Help browser and hovered over some URLs, it appeared that all links using "matlab:web()" opened in a new tab, similar to the "Acknowledgements" URL in the screenshot below:

However, using the "file:///" URL scheme can allow you to open pages in the current tab. Given that your approach involves local files, this method could be suitable.

You can try incorporating this approach in your live script to generate HTML pages that link to your local web pages. I have not seen many File Exchange toolboxes that create help pages directly in the MATLAB browser, but it sounds like a great UX enhancement for your toolbox users.