Depending on what you mean by viable...
The link is to a Grappler Guide in ENWorld. Sadly, the actual builds were not included (yet, but as it was published 3 years ago, it probably won't ever be included).
I will copy-paste the Pros and Cons, then make some analysis on what you are losing for not having spells (spoiler: not thaaat much) and then explain the basics for the build from my perspective.
I have played grappling-focused characters in a few campaigns. They are most useful in early levels. Specially in later levels, the non-spell restriction will make your life even harder. So, answering your body questions: they can be roughly as useful as other classes, depending on the campaign, or completely useless if the DM keeps throwing creatures that shut the build down. In higher levels, its utility decreases considerably, from my experience, and you should probably start focusing more in damaging later on. As a note, my experience is mostly with using EK or multiclassing into a spellcasting class (I love spellcasters, my most played class is Wizard, welp), but I think I have enough experience to understand the implications of not using spells.
- Very hard to interact with. Most monsters are about as good at Athletics/Acrobatics as a first level character.
- Negates mobility. It's amazing how many opponents rely on movement and how many can't do anything once that speed hits 0.
- Makes monsters easier for your party to hit. Grapple them, knock them prone, don't let them stand up, don't let the run around: it's "I hold, you punch" at its finest.
- Cripples offensive capabilities. Grappled monsters already don't get free reign to choose targets or use their weapons. Now give them disadvantage on top of that, or even take their weapons away.
- Combo potential; Grapple a spellcaster in a zone of Silence and no one will hear him tap. Hold two guys in place while your Wizard rains Fireball on their heads. Plant the badguy on the ground as your rogue stabs away. Jump out of a window while holding your targets. The possibilities are endless!
So, you can see from all the pros that it fits a "supportive" character.
- Weak against large numbers of opponents. As in real life (for those martial artists in the audience), grappling is not particularly effective against multiple attackers. You can only meaningfully engage with as many attackers as you have hands, which is often just going to be two.
- Relatively low damage. Most grappling builds don't deal too much damage on their own, and will have to rely on either allies for massive damage output.
- Low mobility. You will often find yourself in situations where you can't dart around the battlefield without releasing your grappled targets back into the fight. This can be a problem in chaotic battles.
- When you get shut down, you get shut down HARD. Come up against a monster that's immune to shove and grapple? Have fun poking him for the rest of the encounter. Run into that one spellcaster who pre-cast Freedom of Movement? Better hope someone has Dispel Magic. Dragon won't land from the sky? Grow wings or throw insults at him.
- Not for the faint of heart: Don't all guides have some ridiculous weakness like this? Seriously though; if you don't want to get in the face of something two size categories larger than you, a frenzied enemy with a greatsword, or a wizard casting damage-maximized evocation spells, this isn't the class for you.
Note that the "low damage by himself" is actually what you want, it seems from your question. The weaknesses number 1 and 4 are the most relevant and harmful ones.
What do you lose for not using spells as a Grappler?
While not becoming inviable by not using spells, Enlarge/Reduce greatly increases your utility by letting you grapple Larger than Large creatures, as well as advantage in the grappling Strength contests.
Other than that, some other utility spells that increase the scenarios where you are actually useful are
- Longstrider gives +10 MS. Not a very huge loss.
- Shield gives you +5 AC as a reaction for one turn. It has somatic components, so if you are grappling two guys, you don't have free hands anyway. Not a huge loss.
- Silence would help you a lot against Spellcasters. Welp. How common are you facing spellcasters and how useful do you want to be? Mage Slayer will help, but it would be better to have some spells together.
- Dispel Magic and Counterspell also would help against spellcasters (check Con #4 above - better than hoping someone has dispel magic, you could have). Also, see above.
I am also not sure on what do you mean by "not relying on magic". This build does not rely on magic, but if you could grab some spells, it would help.
Your main goal is to Grapple + Shove, i.e., grapple the target to 0 MS and then shove him prone. They won't be able to stand up while being grappled.
As you are not getting spells to give you advantage in the Str contests, you should probably go with barbarian to get Rage, which will give you that advantage. The Extra Attack will allow you to Grapple+Shove.
A Fighter also works. Instead of going for EK as I usually do, Battle Master will be your option for nonmagical grappler.
Dipping 1 level into rogue for the Expertise is worth, as you don't have magic support for your grappling.
So, from my experience, this is how you help: you are the tank. If you have lots of Ranged PCs in your party, you will be the one tanking the damage while everyone else is hitting for free. If you have melee PCs, shoving the enemy down will give advantage to everyone in your team and disadvantage for the enemy to hit your party. If he tries to escape from your grapple, he used his action and won't be attacking anyone. You can then grapple him again in the next turn.
I also play a lot of Support Wizard for control. For AoE, the Wizard simply outmatches the Grappler by a large gap. For single target or two targets, the grappler shuts down the foes harder. I had encounters supposed to be Deadly that went easily because the enemy was down and kept failing the contests.
Eventually you will be hitting the enemies, as when you are already grappling someone, there aren't many actions you can take. Instead of hitting, you can Help. Note that Help does not require a free hand.
Overall, sadly, the spellcasters are more useful, specially because they are way more versatile, but that doesn't make this build inviable. As I've mentioned, there are scenarios (single target, sometimes PvP or Party vs NPC Party) where this one outmaches the spellcasters.