@aristotle - absolutely agreed - in fact that's probably the key signal used to establish a brand in terms of SEO.
However, looking at the ways you can achieve that, the work of a branding agency and subsequent professional brand strategy is the best route to take. Any work you do to establish a brand offline (or online in a non-SEO capacity) should naturally result in increase brand searches.
Alternatively if you are putting the same hypothetical budget into SEO (or even broader SEM), you may be able to achieve non-branded rankings (or traffic via PPC, etc), but when you've spent your $20k, will there be an established "brand" leftover that will bring you regular business? In some cases yes, but largely no.
The two approaches aren't mutually exclusive however. For example, if you would take what you recommend in your first post - I would consider that as the second step in the process, where the brand strategy is the first.
Creating a site that has an impact is the objective, but the brand strategy is the plan to achieve that. Just going through that process can lead to new ideas that really bolster a SEO campaign. If you can really pinpoint how you want your brand to be perceived, it makes it much easier to identify sites that might want to link to you or to write copy for a PPC ad for example.
All a branding agency would do is give a project some focus and help pad that out into a solid plan of action. It's acutally mostly common sense stuff that most business owners do any way - they just don't think about it in those terms.
Think of it like this - most web developers "do SEO" to a certain extent. They follow basic web standards (which creates a SEO friendly site) for example. They just don't think about it as being SEO.
In the same light, business owners "do brand strategy" to a certain extent.
The turning point in a project is when the developer decides it's time to invest in professional SEO (i.e. the project has grown to a size to justify it). Similarly, a project can grow to the size where professional brand strategy is useful and justified.