I enjoy growing large zucchini (I prefer them to small ones). So, I can tell you about my experience with the varieties we've grown (whatever they were). My experience is that if you pick them sooner, they'll grow more fruits. However (with the kinds we grew), if you don't, you can still get a few to several large zucchinis on one plant (usually about 2-5). You should be able to pick a few small ones and then grow some large ones. I don't recommend growing the large one first and then picking small ones, if you want small ones.
If you plant an extra-productive variety, you may get more. I hear Dark Star is very productive (and does well in hot, dry areas). Since I first answered the question, I've gained more experience: I grew Dark Star in 2016, and it's a very nice breed. It's very vigorous and productive. It has a mild taste. I grew multiple plants per hole and they still did well. Squash bugs like it, though.
As to how long to leave the zucchini on the plant, if you're saving seeds, I'd say as long as possible, but I really don't know at what point the seeds become viable (although doing a search online, people say they're viable when you can dent the skin with your fingernail, although it seems to me like you could do that with any zucchini: maybe they mean when you can dent it without puncturing it). In 2016, I just left my zucchini on until squash bugs about finished killing the plants (they were full-sized for at least a few weeks before harvest). I kept them in storage like winter squash until January or so when their skin started turning orange, although people say seeds don't develop further after harvest. Then I saved the seeds, which were plentiful and nice and thick. I haven't grown them, but I'd be surprised if they weren't viable (hopefully I can update this after I grow them). The fruit was still suitable for zucchini bread at this point (I did have a box fan running in the storage area to help keep mold away, though).
I just washed them, zapped them for 45 minutes in water with my Z4EX (15 minutes per frequency) as an experimental precaution against disease, and put them in/on a brown paper lunch bag to dry in a room with a box fan going to assist their drying). You don't have to zap them like I did. Some people ferment their seeds (to help clean them and perhaps sift out seeds that aren't viable, and maybe as a precaution against disease), but it doesn't appear to be as common with zucchini as with other things (e.g. tomatoes and watermelon). Washing and drying them should be sufficient for most people. I didn't have trouble cleaning my seeds without fermenting them.
I've saved seeds from a friend's end-of-season, dark green-skinned zucchini before this, which were reported to have been viable (I gave them to a friend to try). I just washed and dried those. That kind had orange flesh due to its maturity. My Dark Star zucchini was yellow inside (a pineapple-type color). Since my Dark Star zucchini were probably more mature than the other ones, I'm guessing whether or not they turn orange inside is based on the breed.
If you grew other C. pepo squash (and possibly some other species), they might have crossed with your zucchini, but new hybrids can have advantages (e.g. productivity), if you're adventurous (but they may be quite different from the mother plant). Crossed plants are hybrids (the first generation is an F1 hybrid).
Please excuse the next part of my answer (which is probably more suited to Seasoned Advice).
If you can't stomach the thought of eating only large zucchini, let me tell you some practical uses for them:
[Not all large zucchini are suited for these purposes. Some have very tough skins and an unpleasant taste. Dark Star and Black seem to be capable of tasting fine for these purposes when large (but it's possibly the soil that makes the difference in taste and such):]
- Slice them (take out the pithy middle) and put them on hamburgers.
- Slice them thin (take out the middle) and put them on pizza.
- Pickle them.
- Make mock apple pie (the zucchini kind).
- Zucchini bread. (Or use them in other veggie breads.)
Really, they're quite versatile. I like to eat them raw, too. They're probably great cooked with onions (not garlic), too.