It seems you already have pretty much the best you can do. As you probably know, Pure Monks are unfortunately not that good grapplers (which even optimally is not a very reliable build).
A few things you should be looking for: A way to get Enlarge/Reduce. Unfortunately, without multiclassing, you can not do that by yourself, so either stock Potions of Growth or ask your buddies to have it for you. This will become necessary in the end game where many enemies are larger than Large.
Can I get advantage on Athletic checks somehow, without party assistance?
The Magic Initiate feat may be useful for you as well, since it's the only way you can get spells with your choices of class. You can get Hex, and give disadvantage to enemy checks on Dex or Str (whichever is more likely that they use to escape your grapple). You can also get Guidance, which will help with the +1d4. Finally, as I mentioned, you should prioritize a way to get Enlarged, which will give you advantage on Grapple, but this does require party help or magic items.
I would not go for this feat if you can rely on any ally to help you a bit - either through Hex or Enlarge or both (with two spellcasters).
You can forego 1 point of AC in Dex and focus on Strength from the beginning, then attack using your Strength instead of your Dex. This will make you more vulnerable, which is probably a problem, but you are also better at grappling.
I would also postpone getting Prodigy to later. At 4th level, +1 in Str modifier gives you +1 in grappling, while expertise will give +2, but Str is more generally useful (for your attacks). From 5 to 7, you will be getting +1 instead of +3, but still you get +1 to attack and +1 to damage (this assumes you take the previous advice of focusing on Strength early on). Then you can get Expertise at 8th level.
At 19th level, I am not sure Dex is what you are looking for. You will certainly be attacking with your 20 Str by then, and Dex only provides +1 to AC. You can get the same by increasing Wisdom instead, and or you can get 19 extra hit points by increasing your Con modifier.
Alternatively, you may also stop at 16 Str, and then focus on Dexterity for both damage and AC. At high levels, you will be getting something like +10 from Expertise, so the difference between +13 or +15 is arguably small, especially if you actually manage to reliably get advantage. This will significantly increase your survivability and should not meaningfully decrease your chances of engaging and maintaining grapple. This will also make you more useful in the scenarios where grappling is impossible (larger enemies, flying enemies). Ultimately, I would go with this choice, to be honest.
Is there a better way to make a monk a better grappler?
I think this is it. Monk is already a very MAD class. You need Dex and Wis for your armor and damage, and you need Con to compensate your d8 Hit Dice. Adding Str as an extra required attribute makes it quite hard to build, and on top of that you are not interested in dipping some levels of multiclass to get useful spells and features such as Rage, Action Surge or Expertise from the Rogue. With all of these limitations, I think this is the best build you can hope for. Everything else will depend on convincing your DM to give you some magic items.
Compendium of Useful Stuff
I just wanted to include some magic items that may be useful, if you have any way to ask for them to your DM. As you already mentioned, Belt of Giant Strength would help a lot with the MAD problem. Items that increase your AC, as you will not be getting as much Dex as you would like otherwise, also help a lot, so Cloak of Protection/Ring of Protection and Ioun Stone of Protection. Alternatively, items that set your other attributes to a high stat, such as Amulet of Health (19 con), or items that increase your stats such as Belt of Dwarvenkind (+2 Con), Tomes and Manuals are also a priority for you.