Correct high altitude operation varies based on whether your engine is turbocharged or not.  I live in Colorado, so I have lots of high density altitude experience.

Most turbocharged engines should be leaned for ground operations, but not for takeoff/climb.  Turbocharged engines usually have altitude-compensated mixture controls and should be full rich at high power regardless of altitude.  These engines also develop 90% to 100% power at all plausible takeoff altitudes.

Naturally aspirated engines are more complex.  They should be leaned for takeoff, but at most density altitudes, should not be leaned to peak.  The higher you are, the less enrichment is needed.  Even in the hottest part of summer, 5000' is too low to take off at peak.  One procedure is to lean to peak during runup (I usually use peak RPM, ignoring the EGT), then enrich from there.  How far to enrich depends somewhat on the plane, but a rule of thumb I use is to enrich proportionally to the altitude, relative to 10000'.  At 10000' I would take off at peak, at sea level full rich, and at 5000' halfway.  At 7000', I would enrich about 30% from peak.  This is true altitude, not density altitude, and assumes the mixture range is not excessively rich (many planes have mixtures that are way too rich - the Piper Arrow I used to rent had a mixture that was so rich that most of the useful settings were leaner than the stop where the idle cutoff is supposed to be!)  In that Arrow I just developed a sense of where the mixture needed to be for typical situations.

Taking off at high altitude too rich is not much better than taking off too lean.  The engine will not develop full power, and "full" power at those altitudes is substantially less than actual full power.  The tree at the end of the runway is no better for your engine than running too hot (but most Lancairs have enough power to take off even when improperly configured).  The famous Stinson accident on Youtube might not have occurred if the pilot had properly leaned the engine.

The prop responds faster than the EGT gauge and at lower power, so setting mixture based on the RPM is usually easier.

If you look at most engine manuals, they will specify a power setting below which peak operation is authorized.  It is easy to see what your maximum power setting for a given takeoff altitude is by looking at the manifold pressure gauge when the engine is off, and dividing it by 30.  (To be fully accurate you would also correct for temperature, which is worth a few percent).  At most plausible takeoff altitudes, the 1" per 1000' rule will apply, so at 5000' you can expect 25" of MP, so your engine will develop about 80-83% of maximum power - too much to takeoff at peak.  At 10000' (suppose you flew to Leadville) you will develop a maximum of about 60-65% power, which for many engines would mean taking off at peak would be acceptable.

Of course, in all cases, you need to enrich to keep CHTs acceptable.  Even if the engine manufacturer authorizes peak operation at a given power setting, cooling is another story.  You might not have enough cooling airflow in a climb, and the fact that the air temperature is well above standard means it doesn't cool as effectively (another way to look at it is the lower density air has less ability to conduct the heat away).

Finally, remember that (in terms of IAS) Vy is lower and Vx is higher when at high altitudes.

The "reference EGT" method is also good, but requires more attention as you have to monitor your EGT during takeoff.

Above about 12000', just about every non-turbo engine can run at peak, as you will be at or below 60% power.  Turbo engines should run LOP when not climbing or in a very shallow climb, and should never run at peak.