Is there a good, satisfying ovo-lacto substitute for meat for recipes like meat & cheese lasagna? Or can I just skip the meat and expect the dish to satisfy without it?
I understand that tofu would add protein - but would it add flavor?
Meat lasagna minus meat will not be as satisfying - it'll be missing both a flavor and texture component. And tofu by itself does not have much flavor, especially compared to the rest of the lasagna.
Substituting for meat is a broad question, as you'd see if you'd tried searching for vegetarian lasagna. If you're determined to make something that's as close to lasagna with meat as possible, then you probably want to use something that's actually intended to be a ground meat substitute. There are various options, but they're generally soy- and gluten-based, with some sort of flavor. This will be good for vegetarians who wish they were eating meat.
But for vegetarians who don't feel the need to pretend they're eating meat, and especially for ones who don't want to feel like they're eating meat, you're asking the wrong question. Don't ask what to replace the meat with. Ask how to make a good vegetarian lasagna-like dish. The flavor of mushrooms works well (including plenty of umami - note that mixing varieties gives the broadest flavor) and provide some nice contrasting texture to the rest of the dish. Peppers and onions also have great flavors, and don't drift too far from the original flavor. I tend to enjoy spinach and eggplant too. Just try searching online for vegetarian (or vegetable) lasagna, and you'll find tons of variations. They're not all for everyone, for various reasons, but there's plenty to look through for something that works for you.
As for tofu... there's already a ton of cheese in a lasagna, which does have protein. Unless you're really worried about protein intake, I don't see focusing on protein as the route to a pleasing dish here.
(And this advice is pretty much all general; it applies to most flavorful casseroles that might normally contain ground meat.)
My favorite meat replacement for lasagna is mushrooms, preferably a mix of the more hearty mushrooms like Shitake, Lion's Mane, Porcini or others. If those aren't available, most stores have "Baby Bella" or "Crimini", which are just the young version of the "big" Portobellas, and a darker variety (with a little more flavor) than the white button mushroom. There are some vegetarian sausage replacements which are satisfying as well (Tofurky brand tend to be pretty good, and well spiced).
Other vegetables go great in lasagna as well - spinach, peppers, onion and summer squashes all go great with it. The big thing you have to do is cook as much of the moisture out as possible - vegetable lasagnas can get "soupy" quick because of the high moisture content of the ingredients. A long sweat (cooking at a temperature just below a saute) with a bit of salt added to draw the moisture out of the vegetables will help. For larger quantities, toss them quickly in a little oil and salt, spread in a single layer on a sheet pan and roast at 425F until they lose most of their liquid and start to brown.
You can also further reduce your sauce by cooking it a bit longer, and parcook your noodles (boil for only half the time) to allow them to absorb more moisture during the baking.
If you want to drop the cheese as well (not necessary for an ovo-lacto vegetarian), then you would be best off ignoring the "melty gooey" portion entirely - vegan cheese does not do the melty thing well. However, some of the Ricotta substitutes are quite good, and play to the strengths of the vegan ingredients. Firm tofu, pressed of water, and then "scrambled" with a little olive oil and salt would do nicely as well to lighten the flavor a bit.
Mushrooms for flavour. Plenty tomato in the ragu sauce. Pep it up with some basil and other herbs. There is no reason that a vegetarian lasagna shouldn't taste good - though it won't taste like a meat lasagna - it can still be satisfying.
Tofu, though nutritionally good, tastes of little (similar to the pasta) and will make for a dull lasagna. In fact, tofu skin could replace the lasagna pasta but what would be the point of that. If you want to cook with tofu, look to Chinese and Japanese cuisine where tofu is a common ingredient or even primary ingredient.
If you really need the texture of the minced meat, try finely chopped broccoli. However, I would say for vegetarian cooking, try not to recreate meat but rather produce a good tasting dish with what you have at hand.
I think spinach ricota lasagna is an excellent replacement for a meat lasagna. Ricota is very rich type of cheese (good amount of proteins) and the taste of the lasagna is amazing!
Finely diced mushrooms (mix of "normal" and shiitake) as suggested above, PLUS finely crumbled extra firm tofu, both well sauteed, PLUS something that adds umami (MSG, maggi seasoning, soy sauces, fermented black beans, doenjang, ... - be careful not to discolor your sauce too much).
Fine TVP pieces or cooked unripe spelt also can help attaining a minced-meat like texture.