How can I resurface concrete sidewalks for easier cleaning?
The pressure washer erodes some of the surface.  The rougher surface is safer for walking (less likely to get slippery when wet), but it also gives mold, algae, and lichen a better foothold.  Once they get established, they embed themselves in pores, micro-cracks, and other imperfections below the surface.  Getting rid of it by pressure washing requires eroding more and more of the surface, which compounds the problem.

Independent of resurfacing, you need to first get rid of the growth.  Chemical treatment is a better long-term solution than pressure washing because it doesn't keep making the surface more susceptible, and some chemicals will help to prevent regrowth.

There are a number of disinfectant cleaners based on benzalkonium chloride, such as "Wet and Forget" or "Spray and Forget".  These are often sold as concentrates that you dilute to about 2% active ingredients.  Those products are pricey, but the same ingredients are used to kill algae in swimming pools, and is much less expensive.  For example:



This is a 30% concentrate that you dilute to 2%, so it goes a long way.  If the problem is mold or algae, you spray the surface to saturate it.  If there is also lichen, you respray the surface a second time while it is still wet from the first spray.

You don't rinse it off or scrub, just let it soak in and dry.  It is very slow acting.  It can take days to weeks to kill the growth.  Weather then washes away the dead material over time (months).  They recommend not pressure washing it because supposedly, residual disinfectant gets redissolved by rain and kills deeper layers of growth.  A treatment is supposed to prevent new growth for a year, so once you get rid of it, repeat a single treatment once a year.

There are lots of other chemicals that will kill it.  For example:


"Cloudy" (sudsing) ammonia diluted 1 cup per gallon of water will kill mold or algae.  Lichen will require 1 part ammonia to 4 parts water and scrubbing.  It can take a week for the dead growth to wash away and it doesn't leave any protection against regrowth.
Chlorine bleach 3% solution and some scrubbing will get rid of it.  It's the most common ingredient in fast-acting mold cleaners.  However, it doesn't leave any protection against regrowth and requires work.  Chlorine bleach will also discolor the concrete.  
Vinegar will kill it but is less effective, will corrode the concrete, and doesn't leave any protection against regrowth.
Borax solution will kill it but requires scrubbing.  It will leave a protective residue, but it isn't clear whether that's good for the concrete.  
Washing soda (sodium carbonate), like borax, will require scrubbing and will leave a protective residue, but again, it isn't clear whether that's good for the concrete.  It's the active ingredient in Concrobium Mold Control (1% solution).  It also isn't clear how fast the borax or washing soda residue will wash away in the rain.  
Oxygen bleach, like OxyClean, is sodium percarbonate, which breaks down into hydrogen peroxide and washing soda.  So it's an expensive alternative to washing soda that adds some weak bleaching action (maximum hydrogen peroxide concentration of 3%, same as over-the-counter solution), but it won't discolor the concrete.  It's the active ingredient in Concrobium Mold Stain Eraser.


As Ecnerwal notes in a comment, don't randomly mix cleaning chemicals.  Some, like ammonia and chlorine bleach, create toxic gas when mixed.  Even when used alone in these concentrations, you should wear protective gear when spraying or handling many of these materials as they can damage or irritate your lungs, eyes, or skin, and at least the chlorine bleach will also do a number on your clothing.

I don't recommend trying to seal the surface.  I'm currently dealing with a previously sealed surface.  The sealant worked great for a number of years.  Nasty stuff is now growing under the sealant, and the sealant blocks all chemical treatments.

As far as resurfacing, you could polish the surface, but that would be dangerous for walking and wouldn't make a big difference at this point as far as growth.  You might be able to bond a new surface layer to the top that contains growth-inhibiting material, but that isn't likely to hold up long-term.

The "best" solution is probably to use chemicals to kill what's there and leave a growth inhibitor.  Then once a year, pressure wash at a low pressure to clean off the surface without eroding more concrete.  Let it thoroughly dry, then re-treat it with a growth inhibitor.