You need a better understanding of the way speakers are set up. No speaker can reproduce the entire range of sound on its own. For this reason you usually see speakers consisting of at least two components: a woofer to produce the low to mid-range sounds, and a tweeter to produce the high sounds. A speaker like this would be referred to as a 2-way speaker. You may also see 3-way speakers, consisting of a woofer, mid-range driver, and a tweeter, or you may even see 4- or 5- way speakers. Though these speakers all consist of multiple "speaker" components, for electrical purposes they only count as "one" speaker.
Speakers, at least the automotive variety, usually come in one of two configurations: coaxial or component. Coaxial speakers have a large woofer with the tweeter mounted right in the center. Take a look at your Alpine 6.5's and you'll see what I mean, they are 2-way coaxial speakers. Component speakers will still have a woofer and a tweeter, but they will be separate from each other and can be mounted in different places. Coaxial speakers are generally easier to install and get good sound out of. Component speakers are supposed to produce better sound than coaxial speakers, but require more effort and must be set up properly to do so. If not installed properly component speakers will likely sound worse.
To understand how these speakers work you also need to understand crossovers. Though each component of a speaker only produces particular sound frequencies, your head unit or amplifier puts out the full range of sound on each speaker channel. Once this full signal reaches the multi-component speaker it must be broken down and distributed to the proper components. This is what a crossover network does. It will take all input below a certain frequency and send it to the woofer, all input above a certain frequency and send it to the tweeter, and so forth (depending on the number of components). Coaxial speakers have crossovers built into them; component speakers have separate crossovers that also must be mounted somewhere.
This is why a normal four-channel soundsystem will have four 2-way coaxial speakers attached; electrically speaking the amp does not see eight speakers, it only sees four. Your six speaker soundbar is actually mis-named, because it is not six separate speakers; what you actually have is two 3-way component speakers. Each set of three only counts as "one" speaker, electrically speaking. I could buy a pair of 3-way coaxial speakers and install them in my factory sound bar and have the same effect; the only difference would be mine would "look" like two speakers, while yours would "look" like six.
This is where your problems begin. You pulled the woofer component out of your 3-way speakers and replaced them with 2-way full-range speakers. The original woofers were part of a crossover network, while your Alpine speakers have built in crossovers of there own. This all gets over my head when we get into crossover setup, but I will bet that no part of that sound bar is working properly anymore. At the very least I would guess that the Alpine speakers are only receiving low-frequency sounds and the built-in tweeters are doing nothing. At the worst I imagine it's possible that the two separate crossover networks are getting some wonky interference from one another. Add that to the fact that VDP probably did not use very good quality crossover or speaker components to start with, and your sound bar probably sounds nowhere near as good as it should.
You were correct in guessing that you should disconnect the smaller speakers, though not for the reason that you thought (again, your amp does not think you have eight speakers connected to it, only four). Just as an FYI, you can certainly connect more than one speaker per channel; however, depending on how they are wired you will change either the current draw or impedance of the load, which is a whole other mess that I do not fully understand or have any interest in getting into. Just know that if you want to wire more than one speaker per channel and they are not components on a crossover network, you must do the math to figure out how to set them up right.
If you disconnect the smaller speakers, you need to make sure you take those crossovers out of the loop as well; each of the Alpine speakers needs a direct connection to a channel from the amplifier. Doing so should get you far better sound. Those coaxial Alpine speakers on an amplifier should perform way better than whatever junk VDP used in their component setups. This would probably the easiest thing for you to do to get better sound right away. If you want to try to get better performance from a 3-way component setup then you will need a new set of properly matched tweeters, mids, and woofers AND new crossovers. Then you may get better sound than properly wiring those 2-way Alpines, but it will be a lot more work and expense.
Another thing to check would be your high pass/low pass filter settings. Filters are basically crossover networks that are built into an amplifier or sometimes head units. Rather than doing the frequency splitting at the speaker end of the line, it does it at the source. If you were using your amp to drive a subwoofer you would turn on the LP (low pass) filter, thus sending only low frequencies on that channel to the amp. If you had a subwoofer and then wanted to only send mid to high tones to your smaller speakers, you could turn on the HP (high pass) filter. Since you should have four full-range speakers connected to your four channel amplifier, check the amplifier and make sure the "Filter" switches are set to "Off". Also, your Alpine amp also has a "Bass EQ" switch which is for use with a subwoofer, make sure that is off as well.
Lastly, you should not have to worry about your amp and speaker wattages. Yes, your speakers are rated for 80 and 55 watts RMS. However, your 300 watt amplifier is actually only rated at 50 watts RMS per channel when used with four channels, so it shouldn't be a problem.