I'd slap some big Adsense on it and enjoy!  

I believe that one off topic page will not hurt anything - especially if the page is an outstanding example of content for its topic.

I know sites that publish on a wide range of content and have fantastic rankings.  These sites were once single topic,  then departed with a single page that struck gold - and then started plowing that field into a hundred pages,  and soon ranking #1 for the single-word root keyword.    They then hit gold on a second topic, and a third.  Now these sites have diverse branches but have found ways to interrelate the topics.

I don't think that Google has a problem with conflation.  You might even make Google think that the two topics are complementary.

their own business name

In almost every instance page one organic position should be possible in a reasonable amount of time... and if they are a local business the time required for them to be in the #1 organic position should be quite short.

keywords they want to rank for

Some keywords have brutal competition and in many cases page 1 organic rankings will either be impossible for the business or so costly that acquiring them will not make economic sense.   The problem that many clients face is that they have hired an SEO who doesn't really know what is required to attain the desired ranking, charges the customer a certain amount per month, and the SEO does not have the ability to pull of the desired ranking even if the budget was 5x as much.  In these cases the client is left in anticipation that goes unfulfilled.

So, the SEO needs to:  1) know what the client wants; 2) determine the business reward for acquiring it; then, 3) explain the probability, the cost, the timeframe for acquiring it (making disclaimers that aggressive efforts of other businessess in the space can move all of these goalposts at anytime).  Then the client is fully informed and they can search search search with little concern to the SEO that they are looking for to deliver results.

Dangerous?

I doubt it.

I have a site full of this and it is hard to beat in my SERPs.

Saves time in my opinion.  Smart, if the title and the H1 make sense.

Lots of content management systems produce this automatically - and match the URL to boot!

Added: (I do have lots of pages where the title tag and the H1 differ.  For those pages the title tag was written to elicit clicks and the H1 was simply the title of the article.)

I have 301 redirects on my sites.   Every one that I have done is still out there in htaccess.  I am not taking chances on how search engines handle these.

Other than deleting useless pages, I rarely change URLs.  If I don't change URLs I don't have to worry about this stuff.

I have not emailed other sites to change the URL in their links.  I have only changed the URLs on my own sites.  I would worry that asking someone to edit a link might result in a loss... so I am happy with the redirected link.

I have a little different point of view.

If you are a very smart and very motivated person I would join SEOmoz for a few months and  get all of the learning possible here.  EVERYTHING.  At the same time I would start a website and start experimenting with what I have learned.   (Do not do  this on a site that a business or employees depend upon!)

Ask lots of questions here in Q&A.  Not general questions.  Ask very specific questions about what you don't understand.

Then, after you have learned a lot and have a site working to the best of your ability, hire an SEO for consultations on your site.  Have him/her look at what you have done and recommend on how it can be better.  Keep working on that site.

I believe that this study, work, and consultation will yield really good results and give feedback on a practical project.  And since you have skin in the game you will work harder and see practical benefit than simply working through a course that might not be directly relecant.

The average Joe Schmoe who has a website is going to type the title of his page into both the <title>and <h1>.</p> <p>I can't imagine any search engine deducting points for it.... and with 199 other factors in the rankings if this WAS in there the deduction for it would be really really low.</p> <p>Still... I think that a person can safely bet a month's pay that matching title and H1 will not make him rank #2 when everything else ranks him #1.</p> <p>Also, I can't imagine search engines placing this "invisible tripwire for SEOs" in their evaluation that is going to be automatically triggered by some of the most commonly used content management systems.</p></title>

I own several keyword domains.  Some of them have top rankings for their exact match query and some of them don't.   There is no special formula for beating them.  Just compete against them as you would any other domain.

If you ask me the bigger problem is google giving easy top rankings to weak content on ehow, about, wikipedia and other powerful sites.

How hard is it to get these very annoyingly favoured domains off 1st?

This really seems to bother you.  But if you turn that around you would consider them to be a huge asset.  So, maybe you should just go out and buy one.  Find the guy who owns one and ask "what will it take for you to sell it to me?"... or hire a pro to do that for you.   I've done it a few times and am happy with most of the results.  They seem to produce a higher conversion rate too.