Sticking to one playstyle

I had this going on in my mind for quite some time now, so Im gonna ask this - is sticking to one certain playstyle, say support, good for the long run?
I'm not saying I'm sticking to just one hero, but instead get really familiar with all the support heroes while maintaining that babysitter playstyle ('cause tbh I enjoy that playstyle for whatever reason).

And while we are at it, what are some good support heroes apart from Lich, Maiden and Omni?

its good. it makes you very familiar to playstyls of one kind-making you much better at that than others.

other good supports : Io, Ogre Magi, Dazzle, Shadow Shaman, Sand King.

But of course it makes you less versatile to situations

Support:
Dazzle, pretty fun hero with quite the impact
Abaddon good laning but lacks long run game impact
Shadow Demon super strong but a bit complicated
Io awesome guy but VERY HIGH SKILL CAP

Sticking to one play style during your learning phase is very helpful. By playing just one type of hero (not just one hero, but a set of heroes with the same role), you can not only master the play style, but also your general game sense. You are more comfortable playing the heroes you are most familiar with, so you can learn more advanced game mechanics, like positioning, map awareness and overall game sense, while playing them. Basically, you become a better player overall compared to someone that just tries out all the heroes or tries to learn multiple play styles at once.

It's probably for the best while you're learning the game, but once you have 50 games with a couple of Supports, you'll definitely be ready to start learning other playstyles. You'll never be a great at any Role until you've learnt the basics to them all, but focusing on getting the hang of one at a time makes it easier to learn, bit by bit.

Supports in general, or just Babysitters?

Easy to learn:-

Lion - strong Disables and can replenish his own Mana with Mana Drain, Finger of Death is fun as well <}3
Lich - can't do much other than sit in Lane spamming Sacrifice and Frost Blast early on, but Chain Frost can win Teamfights if used properly
Crystal Maiden - not that strong in high skill games, but a point in Arcane Aura at lvl3 can helpful if you can't control your Mana very well.

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A bit more difficult:-

Vengeful Spirit - a strong Roamer, don't pick this Hero to babysit. Always use Nether Swap carefully, sometimes its needed to save an ally, sometimes it's better to Initiate a Gank/Teamfight (using her Ult properly is probably the hardest part of playing her)
Skywrath Mage - strong Harrass with Arcane Bolt and Maxing Arcane Seal at lvl7 can make sure that Primal Split, Teleportation, Sunder, etc. don't happen.
Shadow Shaman - has some serious Mana problems and isn't a strong Babysitter or Jungler and not the greatest Roamer, but once you've got the hang of getting Exp and farm on him, you'll be taking Towers in no time.

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Advanced:-

Leshrac - usually played as a Core these days, but still works as a Support in the right lineup, with no set Skill Build, you'll need to learn to adapt each and every game before you get the most out the blue magical ponycorn
Bane - learning to use Nightmare effectively is only the first trial to learning this Hero. Prioritising the correct spells on the correct Heroes in the correct order at the correct times in the middle of a Teamfight is the real challenge.
Shadow Demon - the "Swiss-Army Knife" of Supports, you can save allies, set up Ganks and...Disrupt Teamfights...and you can also screw your Team over depending on how you use Disruption

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Micro intensive:-

Visage - a strong hero in most lineups, out of all the Supports that can be Micro'd, he's the easiest though, just TAB and Q {x3}, CTRL + RIGHT CLICK and {whatever button selects your hero} + Q/W/ITEM.
Chen - remembering which Creep you have assigned to which button is harder than it sounds at first...unless you play RTSs 8{3
Meepo - ARE YOU READY TO FAIL?!

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With friends only:-

Io - probably the Hero I least want to see picked up in a Pub when I'm Laning...

if you are new to the game, sticking to one playstyle/role is completely fine, altho id say that support is one of the hardest to learn (if you havent that much experience as a player).

Xyrus' list is a very good guideline, but id avoid picking Leshrac, Lina, Sand King or Keeper of the Light if you are the solo support, since those heroes are still very item dependent and tend to become food for the enemy team if they dont get their "core" items fast.

Isn't having such a limited approach a bad thing, however?

For instance: I love playing as Techies, but if I see Keeper of the Light and Night Stalker in the same lane, then I may be in a situation where how I WANT to play the offlane is not the good idea. Wouldn't knowing 1-2 "plan B" heroes be a good idea?

I personally suffer from this a lot.

True, once you learn to spot potential problems, you can learn to adapt your playstyle. This isn't possible for newer players though.

Rather than failing miserably at something they have no idea how to do, they should stick with 1 playstyle in the beginning until they can succeed with it. They may only succeed in certain circumstances, but at least they'll find some success and enjoyment out of it. Before long, they'll become comfortable with it, and gradually realise its limitations. They can then look into other Roles/Item Builds/Heroes etc. and start increasing the breadth of their plays.

Really? I'd actually recommend playing support to familiarize oneself with the game. This may not have been as true a year ago, but after 6.82 with a teams new dependence on late game carries I'd say that support is a good way to go. This also helps you to master skills like map awareness and warding without being TOO valuable to the team (even though supports are everything, they are not recognized as such in sub-4k mmr). I get what you said about those item dependent ones, but an easy support like CM or Lion is well suited for a beginning player.

To OP:
I recommend the supports you mentioned for a beginning player, but branching out later.

Other good supports? I'd say Ogre Magi, Treant Protector, Io (when paired with certain carries).

I don't think there's really a right way and wrong way of doing these things - find something you enjoy, and gradually branch out into other areas.

Support naturally opens up understanding of other roles...play a ganking/roaming support...learn more about ganking. Big team fight ability? Initiators. What worked against you well in lane, why couldn't you kill that guy? Offlaning. Semi-jungling...jungling. Even carry - why does supporting matter? How do you work well with your supports?

For more relatively easy supports - try Lion, Ogre Magi. Jakiro can be a good introduction to skillshots (without them defining your entire success) later on, and Venomancer is pretty useful and not too difficult to play at a basic level. Dazzle is a little more complicated than he first appears, but fun and not too difficult to start with.

If you want to look at teamfighting more, try Warlock, Sand King or Witch Doctor. Fun with ults? Ancient Apparition - although be aware he requires some extra co-ordination with your team.

Some of the other ones require more advanced knowledge, micro or skillsets, I'd avoid them for now.