Why such a restricted set of heroes are advised Hand of Midas?

Saving up for and buying a Hand of Midas weakens your early game to allow you a more explosive late game. The time that you use saving up for a Hand of Midas is time you are not spending buying other items; likewise it's not just the time spent saving up for and buying it, but you also need to (in a sense) use it a certain number of times equal to what you spent to buy it before you even start getting value out of it. It pushes you towards the late game.

So how well does your hero work in the late game?

Is the time that you are spending building a Hand of Midas time that you could be using to make early plays and gain significant advantage?

That's why for such a long time the pool was limited to late-late game heroes, where the Hand of Midas accelerated their farm and allowed them to come online earlier; whereas for heroes that come online early; the Hand of Midas was not so beneficial.

If every hero on your team committed to building a Hand of Midas you would have a very weak early game (first 1-10 minutes) and the enemy would capitalize on this, build some cheap items and push down your towers before you could do anything about it. You might have your Hand of Midas but it wouldn't make a difference if the enemy could overpower you and win by 20 minutes.

Secondly, does your particular hero even need the gold gain?

What are you doing with it? As a support are you really struggling with keeping wards up and finishing one core item like a Mekansm? Building the Hand of Midas is pushing that Mekansm back and the later you get the Mekansm, the less use you will get out of it. It's not accelerating you in this instance; it's just giving you some sustain and a better ability to scale into the late game.

That being said there is such a thing as a catch up Hand of Midas. When your early game opponent is snowballing some and you really need to accelerate your farm to turn it around.

It's not to say it's ever a bad item on any hero, it's just all about timing

Is picking it up really advantageous for your team if your strength is in winning early and you are pushing the game back by doing so.

When people get Hand of Midas on a support, it's usually more for the bonus XP (the creep you kill gives 2.5x the normal XP) than the gold. That's because in high level games, supports spend a lot of time pulling or roaming and often end up under-levelled. I don't think getting Hand of Midas on a support is really justified for the gold alone. I certainly wouldn't do it as a relatively new player. Even in pro games, the Hand of Midas-on-everything meta that was so big recently seems to have tailed off.

Big ticket items like Scythe of Vyse are really a luxury on supports

Aim for them if you're getting tons of gold from kills, but otherwise don't worry about it. I wouldn't say getting a Hand of Midas to help you afford one is worth doing most of the time. Focus on building smaller items that will enhance your survivability or provide some more benefit to the team. Getting an early Bracer and then later building it into Drum of Endurance is a great example - Drum of Endurance give you great stats (almost the same as an ultimate orb), move speed, and a great aura too. Best of all, they're really easy to build. Force Staffs are also easily built, and are amazing even if you don't need one yourself - you can use it to save an ally who's getting ganked, or to force an enemy away from their team for an easy kill. Use it sensibly, and it's one of the most versatile items in the game!

If you want the disable and mana regen of a Scythe of Vyse but can't afford one, consider building an Orchid Malevolence orEul's Scepter instead. Orchid Malevolence gives you similar benefits to the Scythe of Vyse, except with a long duration silence (with bonus damage) instead of the hex. It's also really easy to build - no secret shop, and no component over 900g. Eul's Scepter can be used either to save yourself (since it makes you invulnerable) or to take an enemy out of the fight, while still giving you the int and mana regen plus some bonus move speed. It's also about half the price of a Scythe of Vyse.

The point is, there's almost always something you could be building instead of that support Hand of Midas that would let you be more effective, earlier. After all, a big part of your job is to dominate the early game so that your carries can dominate the late!