This is an excellent question, John. It's the quintessential question that anyone who's interested in webcraft would want to know: how do you use a website for 'blogging?
This question, however, begs another even more important question: what does 'blogging mean?
I'm sure there are many different opinions about what 'blogging means, just as I'm sure there are many different types of beverages one may consume, or products one uses in life. It depends on what you like and how you want to "do" it.
To me, 'blogging means writing: the writing may be plain text, or text with images, or text with other forms of multimedia (e.g. audio or video files). Text can be written in all different kinds of ways: For those who are inclined towards Wordpress, the terminology used they use is 'blog post. The analogue in Joomla! is what we call an article.
Keeping to the Joomla! terminology, articles can be organised or classified by the site owner and/or article author(s) is all kinds of ways. You could, for example, have articles organised into categories for food recipes, art works, science, macrame working, space travel ...
No problem there, is there? Furthermore, these categories can be sub-divided. Even then, the categorisation is not restrictive to using one category for a bunch of articles. What if you wrote an article about cooking chateabriand in space—I know, that's kind of travelling where no-one has travelled before. I'm just saying, what if? You can use J! tags—a feature that has been part of J! since 2013 (around the same time that J! 2.5 was retired from service)—and tag your food recipe article with "space travel" and "science" and "comic relief" ... or whatever else takes your fancy.
There are endless possibilities there. J! has these 'blogging facilities built in and there's no reason to shop around or zip off to Wordpress, Qix or some other whatchamacallit.
I could write—or should I describe the activity as "'blog"?—about how to 'blog in Joomla! for ages. It's one of the main reasons I use J!.
Lastly, John, if I may: J! 2.5.28 is really ancient. You're right about it not being a good fit for mobile devices. You're right about it being out of date. However, it's not a big deal to migrate from J! 2.5.28 to J! J3.x. If you're in any doubt about that, get in touch with me (somehow). We could have a chat (offline from this forum) if you have the time; I like to do these kinds of things on Skype even if I don't make a red cent from doing it.
I had a [very] quick look at your website, John. I'm sorry to say that I was totally lost as to what the site is about and what I could do with it. Keep in mind that the average attention span of the average user is less than 2 seconds, you have 2 seconds to impress newcomers to your website. I think that you could probably do a lot more with a lot less on the home page (that's just my opinion).
Anyway, good luck. It's not that difficult ... and we have all the time in the world to work on projects like these, haven't we?
Cheers