To be frank, it sounds to me like you need a new web designer. An experienced designer knows that the right typeface often comes with a fee. A less experienced web designer, who may be accustomed only to open source software, may be unaware of this and assume that all software and assets should be free. Such a designer is overlooking the fact that type designers are creative professionals just like themselves, and deserve to be paid for their work.
Google Fonts is an unusual case because most of the type designers represented received a nominal advance fee for including their typefaces in the library. But many other type designers (the majority of the field, in fact) prefer to be paid on a royalty basis when their fonts are licensed, whether it’s via a retailer like MyFonts or directly from their own site. The result is that most of the high quality, interesting, and uncommon typefaces are not found at Google Fonts. There’s no doubt GF is a handy service for some everyday uses, but because the fonts are so common on the web it’s not what I’d use for most corporate identity projects, or anything that requires typographic distinction.
Find the best typeface that suits your needs and license as many styles that you anticipate using. Sometimes you only need a couple styles, and sometimes a full family will be a better value than individual fonts. For many projects, the type does a lot of the work in a design, so doing your research (or a designer who does their research) to find the right fonts is well worth the cost.
It’s also possible that your web designer is only familiar with hosted webfont services (like Google Fonts, Typekit, or Webtype) and is unaware that many webfonts can be self-hosted. This means you license the fonts, download the files, and then host them on your server just like other assets, such as images.
It could be that the designer is using a template system that requires Google Fonts and does not allow self-hosted webfonts. In that case, I hope you’re getting a good deal for their services.