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How about a Reverse Undergravel Filter?
Yes, sir, seahorses can certainly be kept very successfully in a simple setup such as that, as long as you are aware of the limitations of such systems. For example, just as you suggested, the filtration system can be as basic as a set of well-maintained undergravels (preferably the new reverse flow designs) that covers the bottom of the tank completely. I know undergravel filters are considered old-fashioned technology nowadays, but they are inexpensive, utterly reliable and foolproof (no moving parts), easy to install, require no modification whatsoever, and work extremely well for seahorses within their limitations. An inexpensive diaphragm air pump will operate the filter and provide all the aeration you need, or you can upgrade to powerheads for greater efficiency and extra water movement.
For the substrate with your undergravel filters, use a coarse bed of good calcareous aquarium gravel such as dolomite, aragonite, or crushed oyster shell 2-3 inches deep, since the buffering ability of such substrates will help maintain good pH.
It is a good idea to supplement the undergravels with an inexpensive hang-on-back filter or canister to provide better circulation and accommodate chemical filtration media. This is a very simple, inexpensive aquarium that's extremely easy and economical to set up and operate, yet it can be very successful if used within its limitations. For instance, undergravel filters are notorious nitrate factories and the hobbyist must take measures to compensate for this fact. This simple system relies totally on water changes to control nitrates. There is no live rock or live sand bed to provide denitrification, no algal filter or denitrator in a sump, and no protein skimmer to remove organics before they enter the nitrogen cycle. This limits the carrying capacity of the tank and makes an accelerated maintenance schedule and more frequent water changes an absolute necessity. For this reason, reverse flow undergravels often work best with seahorses rather than standard UGs; they help prevent detritus from accumulating in the gravel bed.
I recommend weekly water changes of a least 25% for such a system. Use a gravel washer to clean a different portion of the gravel bed (no more than 25%) each week and keep the tank under stocked. If you are diligent about aquarium maintenance, perform water changes religiously, and limit yourself to fewer seahorses that you feed carefully, you will find that a simple system featuring undergravel filters can be very successful. But if you are negligent with regard to maintenance, skimp on water changes, or tend to overcrowd or overfeed your tanks, this system will be very unforgiving. An aquarist with your background can do better than simple undergravels, sir.