I prefer cubed myself, about 1/2 inch in size. That's a finished size so cut it larger to start. It fits on the spoon well and lets some of the rest of the chili be present too. Except for the Mexican chorizo which turns into a bright red puddle of goo when released from its casing. Excellently flavored goo though.
It sounds to me like you're not cooking the cubed meat long enough or low enough. Chili is more stew than soup.
Another good option is to use some meat from a ropa vieja (another stewed meat) which is a tender shred, neither ground nor cubed.
Because chili is best with a longish slow simmer, you need to add ingredients at the right time in the cooking so they all come to completion at the same time. If using the ropa vieja, then it's already cooked tender and flavorful so it should be added late so it doesn't overcook.
If you cook the beans the whole time, they'll turn to mush or break up. That's not all bad, but it should be applied properly. Cook the beans in a flavorful way so that the bean broth can be added to the chili. But also include some beans in the main pot early to break down and texturize the chili. Add the rest of the beans in the last third or so of cooking time so they are pleasantly firm and distinct.
While the beans are doing their thing, prep the vegie base (onions, garlic, chiles, tomatoes if using) with a nice saute and let that simmer along developing flavors.
Also sear off the meat and set aside to add at the right time. If it's a real tough cut of beef, add it early so it can simmer it's way to tenderness. I prefer a mix of meats, some beef, some pork and some chorizo. Each meat brings it's own flavor and texture to the chili.
Nor should every ingredient taste the same in the mouth. It's GOOD that the meat doesn't taste like lime. It should taste like what it is accented with the broth and whatever else is on the spoon. The lime flavor, for example, will cook out of the chili if added anywhere but at the end of cooking. So it's not surprising that the meat doesn't taste like lime.
As to the spicing, Paul Prudhomme likes to add his spices in layers. A spice mix will have a different flavor at different durations of cooking. By adding the mix a little at a time throughout the cooking time, you can bring each of these different flavors of the same spice to the finished dish yielding a more nuanced flavor than if you add it all at once. TASTE as you go so you can recognize these differences and learn what you like. You may end up using a different ratio of spices, some more,some less when cooked this way. So bloom some in the oil when you saute the vegies, and some in the beans as they cook, and as you simmer and combine the different ingredients.
Phil