This is an excellent question. Common knowledge would seem to indicate that there is no problem with eating cooked matza since you definitely can't use it for the seder and that Mezonot-foods seem to be preferred to other non-bread options for the Third Sabbath Meal generally. So why was this not the obvious solution for Erev Pesach (EP)? Why was no one doing this and instead eating meat and fruits? To defend all those people, here are a few ideas.
- Crushed cooked matza may have been prohibited by a different custom.
Rama (471:2) quotes a custom (of which he disapproves, but apparently existed) that forbids crushing matza on EP. A different custom prohibits porridge-like foods on Shabbat EP (444:3) lest it be hard to clean the pots. In principle neither of these customs need apply to cooking matza before Shabbat EP for use on Shabbat, since the crushing is happening earlier and the pot here doesn't need to be cleaned, but perhaps this explains why people refrained from utilizing your suggestion. (Matza balls are not a porridge but it's not unusual to find undercooked matza in the center of a dense ball, so from that perspective a porridge would have been preferred.)
- Perhaps cooked matza is still forbidden on EP.
Is it so obvious that cooked matza is permitted on EP? Just because the reason of the prohibition may not apply perfectly? Would anyone think you can eat Pat Akum if you've dipped it in your soup so it's no longer technically proper bread?
One conception of such a prohibition is that the prohibition on eating matza on EP was always intended to include matza-like foods and not only foods that could literally be used for the mitzva at the seder. Indeed, were this not the case, we could find much better solutions for our Shabbat meal than cooked matza, like matza baked with the intention of not using it at the seder since matza for the seder must be made "lishmah". (Maharsha (Pesachim 99b) explicitly rejects that option and R' Shlomo Kluger (HaElef 322) extends his argument to cooked matza.) Indeed so many Rishonim say explicitly that there is no Pat (bread/matza/hamotzi) that can be used on EP afternoon except matza ashira for those who eat it. Note as well that Rabbenu Tam's allowance of matza ashira on EP is quoted in early sources almost as a novelty. Did no one think of that before Rabbbenu Tam? Perhaps that too was close enough to have been inappropriate for consumption so close to the seder itself. This is indeed how the Vilna Gaon (commentary to 444:1) interprets the Rambam's position. See too Ateret Zekeinim OC 444:1 that some prohibit matza ashira on EP.
Another conception of such a prohibition is that the prohibition on eating matza on EP indeed only applies to bona fide seder-worthy matza, but once prohibited the food can't lose its prohibition. We see this idea in two ways. First, Rama (471:2) writes that crushed up matza mixed with egg or honey can't be eaten on EP because it doesn't convert to become matza ashira after it was baked kosher originally. Many, many later authorities (eg. Meshivat Nafesh OC 4) struggled with this rule since it seems obvious that such a cake would not be considered bread by any rule. Arguably, the prohibition is such that once it attaches to an object it doesn't leave. Secondly, some (eg. Rav Pealim 3:27 quoting Beit David 248) rule that matza cooked on EP remains prohibited for this reason since the prohibition already took effect. While that won't matter for EP on Shabbat since everything must be cooked before Shabbat, it's not hard to imagine an extension to that position which assumes the status of matza takes effect even before the prohibition is relevant, not unlike how chametz before pesach is considered by many to be a prohibited substance regarding kashering vessels (OC 451:4). See too Yehuda Ya'aleh YD 4, Beit David OC 247, Yosef Ometz 729, and Da'at Torah (Maharsham) OC 444:1.
Finally, while cooked matza indeed can't be used for the mitzva at night, bread that was cooked and then baked is still fully bread (like a bagel, OC 168:14) so arguably cooked matza is still potentially usable at night and hence should remain prohibited.
- Perhaps meat is actually better than cooked matza crumbs
The opinion that allows not using bread for the third meal allows the meal to be fulfilled with types of food called "Targima" (Tur OC 291). Targima foods show up in three halachic contexts: The Talmud (27a) tells us that Targima can be used for the minimum number of meals in a Sukkah (according to the opinion that there is such a minimum) and [arguably] by extension the minimum number of meals on Shabbat, by implication we see Targima is obligated to be consumed in a Sukkah, and the Talmud (Pesachim 107b) tells us Targima foods are not subject to the prohibition of eating substantive food on EP late in the afternoon.
There is a three way dispute what counts as Targima foods: 1) cooked grains (תבשיל דגן), 2) meat, cheese, legumes and things that are eaten with bread (מלפפין הפת), 3) fruits. Most Rishonim hold of opinion 2, while some hold of 3 and a few hold of 1. (The main proponent of 1 is the Rosh and thereby his son the Tur and they base themselves off a variant version of a Tosefta which is not found in most manuscripts or our printed editions. 2 and 3 are commonly known as the positions of Rabbenu Peretz and Maharam respectively.)
(Note cooked grains are not the same as cakes/crackers: the latter is a type of snack bread that can be considered bread when eaten as a meal; whether or not cakes/crackers can count for the third Sabbath meal as bread without establishing them as a full meal with the blessings Hamotzi/Hamazon is a dispute among Rishonim, but is irrelevant to our question which is about fulfilling the third meal not through bread. Don't get confused by the similar blessing of Mezonot.)
I listed them in that order because it's likely those who hold 2 or 3 would agree that the items before theirs would be at least as sufficiently substantive as their position. Thus, for the third Sabbath meal, it's best to use definition 1 since all would seemingly agree it qualifies as Targima. This is seemingly why the Shulchan Aruch lists out all the options in that order in his discussion of the third Sabbath meal (OC 291:5). For eating EP late in the afternoon though the stricter opinion is 3 since if 3 is Targima and permitted, foods of type 1 and 2 would plausibly be prohibited as too substantive.
Thus, for the Rama who is generally strict (OC 639:3) about eating EP afternoon even after 6 hours not just 9 hours, he has little reason to risk eating matza balls to accommodate a minority opinion about what constitutes Targima (which is itself relying on an extreme minority opinion that Targima counts at all for the third Shabbat meal) at the expense of violating a prohibition of eating that same food at that time. Even for those who aren't strict from 6 hours on, the spirit of the prohibition clearly dictates against eating heavy cooked matza balls in the afternoon, and that may be enough to dispense with being strict for this minority opinion (so suggests Arukh HaShulchan 444:5).
Certainly it's much easier to argue for the permissive side here to allow eating matza balls and we are talking about "only" a rabbinic prohibition so it's hard to argue too strongly to be strict. But also realize that many great people in great communities did not think that was necessary and it isn't 100% clear it's even permitted. Even if the reason to miss out on the third Shabbat meal that week is just customary prohibitions on certain permitted foods (a weaker claim than those who claim there is no command to eat a third meal), Jewish law clearly (OC 288:2-4) exempts one who is practically unable to eat a Shabbat meal from doing so, such as one with a medical condition rendering such consumption uncomfortable or due to a voluntary fast. There is little reason to worry that the obligation to eat a third meal should force one to eat traditionally proscribed food.
You asked about matza balls, but really the stronger question, mentioned by R' Akiva Eiger 471:2, is from cooked whole matza since it is still Hamotzi. Put two sheets of matza in boiling chicken soup for a few minutes on Friday, take them out whole, and use them as Lechem Mishneh on Shabbat afternoon. This is indeed Rav Ovadia Yosef's recommendation (Yechavveh Daat 1:91, Yabia Omer 6:39). The Targima discussion seemingly wouldn't apply here, so the only excuse we could have to not do that is either accept the above suggestions of a more expansive EP prohibition on matza, a much more reasonable argument for whole sheets of matza which do very much resemble regular matza (see Chiddushei Gr"i Pesachim 56 and Chokhmat Shelomo OC 471 that the prohibition is indeed linked to a status as a kind of bread), or question (as does Da'at Torah (Maharsham) OC 444:1 and Gan HaMelekh 64) to what extent whole boiled matza remains Hamotzi when the "matza" is a modern matza-cracker instead of a classic "soft" matza.