Content is king and always will be. Get that right with 'default-ish' typography (and whitespace) and you are good to go. To convey your message put effort into getting that content into sensible length sentences with sensible headings and 'p' tags (not rocket science). That is it, job done.
In my professional life I have worked with many people that call themselves designers and 'know better than me' about typography. These people really do not care one iota about actual content. They will give me PDF 'designs' signed off as the ultimate in design (according to them) with strange things done to fonts, e.g. extra spacing between characters that has to be replicated. They will specify one of those useless web fonts that is only going to be licensed on the live site and not my development box. They will place 'lorem boring ipsum' in the page instead of the client's content. They will make extra sure that the text will not be easy to read by anyone on a Windows PC that happens to have a high-glare screen. Accessibility standards, e.g. contrast ratio and text size really will not matter to them, for some reason they know best.
To make matters even worse, these charlatan designers will not even work in the medium of the web. They have some useless Adobe desktop publishing tool from the 1990s to work with and do not ever have the 'inspect' window open in a browser, trying things out on live content. It always has to be baked in with some useless amount of 'point size' and 'dpi' settings in their clumsy Adobe tool. I feel like they are still on 'Duplo' Lego when they should be on 'Technik' Lego (where the real fun is). Their little flourishes such as changing the letter spacing are usually an insult to the original font designer, but that is something they feel compelled to do, much like how a Betty Crockett Cake Mix deliberately has 'add egg' so that the person baking it can feel they have actually had some input into things.
When working with charlatan designers and a 'big up front design' waterfall approach (typical of small web design agencies) the CSS is rarely elegant and neat. Every page is a horrid mess of fixes to get something approximate to the designs signed off. You cannot have just the one definition for H1 (or p, or li or anything else basic), each div on each page will have a smorgasbord of padding, margin, position and other monsters just to get the thing positioned as per some dufus PDF. Normally this becomes a specialist task, needing a whole team of 'frontend developers' to get this botched text looking sufficiently illegible. This brings in another layer of people who don't care about things like page load time and flashes of un-whatever content.
Yet, out there, there are plenty of sites that have not been shot in the foot by charlatan designers and their useless design processes. They have had the sense to use things like Bootstrap or Material, where it all works just fine, no useless Adobe products needed. I would say that these new design approaches are leading the way, enabling web sites to be content and accessibility focused with typography that is far better than what the typography obsessed 'lorem ipsum designers' come up with.
The 'art' of web typography is evolving for the better, the only thing that has been holding it back are those that think they know best about fonts and believe the only way to design stuff is with 1990's desktop publishing tools. Luckily they are a dying breed as are the web agencies they work for. Lets keep things content focused and leave actual design ('how it works') to the professionals rather than those that think they know best but actually have not engaged in learning the basics of CSS.
