Yes, and no. It is absolutely true that the training in many Aikido schools fails to train the strength of intention required to deal with folks who train in arts that have competition. This is a very serious complaint against the way that many folks train in our art. If we had competition and that was how people could judge the capabilities of the folks they wished to train with, there would be far fewer Aikido teachers around.
That said, there is a problem with your thinking here. You are trying to compare an art that is one of the more subtle arts being widely taught in the world today. You are upset that you can't handle folks that have trained far less than you have in these other styles. Well, the answer is... of course you can't! Aikido takes far longer to learn correctly than these other styles. In BJJ you can start to get good solid results in a very short time. If you train your ass off you can get quite good in a few years. You can't do that in Aikido.
If you are trying to adjust your Aikido to work (in the short run) against a group of folks who do a sport martial arts, you will ineviatbly focus on developing strong technique and strong center. This will take you to a certainlevel but you will not be able to go beyond that. Many people take this path because they can see the results in the short run.
But Aikido training is fundamentally about reprogramming the mind and the body to react to conflict by relaxing and ridding ones body and mind of tension. You will not do this by trying to pit your Aikido against trained grapplers in the early stages of your Aikido. There is a reason that Aikido training, Daito Ryu training, Yanagi Ryu training, etc is done non-competitively. Training is designed to systematically reprogram your responses and to imprint various motor skills that work in accordance with the principles contained in the art. You won't figure these out by adapting your Aikido to another fighting system. When your training has taken you to a higher level it is useful to try it out against folks who don't know any Aikido and will therefore respond according to whatever their training has been. If your Aikido training has been proceeding in the correct manner, you will find that it doesn't matter but that's a very, long time in coming.
Just do the BJJ training. Do whatever else you want as well. Find an Aikido teacher whose Aikido is of the calibre that can take you to a high level but then don't worry about whether it "works" for quite a while (this may actually be the hard part; don't confuse the physical, violent Aikido teacher as the one who is the strongest martial artist. Look for someone who has strong spirit but is soft. Make sure that he can handle anythiong his ukes dish out and that it's not a matter of collusion; this usually means you need to put your hands on them yourself) I don't mean whether it works at all... it darn well should work against other Aikido people who are training using the same set of assumptions. So over responsive ukemi, wimpy attacks, etc have hot to go. Good solid training in the art.
Get as much experience as you can in other arts... Virtually all of the great Aikido practitioners had extensive backgrounds inother arts before they started Aikido. BJJ is great stuff. But don't try to confuse them at the beginning / intermediate stages of your traiing or you will fail to find the essential principles of aiki.
Remember, Aikido, at least in my opinion, is not an empty hand style. It is a style whose logic assumes that both parties are armed. Atemi to vital points is inherent in the system, although in normal practice these are implicit rather than explicit. Thes atemi are precisely the ones that one would never do in a sport martial art. The lack of weapons and the lack of really dangerous and injurious atemi in the sport martial arts allow for a certain way of doing things. If you adapt your system to their assumptions, their system would always work better.
Just make sure you find the right Aikido teacher. This can be very difficult. There is lots of Aikido out there. There is only a very little really good Aikido out there. Aikido is a system that one does for quite a long time while things don't really work. It takes patience. But if you start with a teacher whose Aikido doesn't really work... well, you will never get to a point at which your own Aikido will work. If there is no one whose Aikido is solid to train with, it would be better to train in an art in which the teacher has taken his training to a deeper level. Training in any art on a deep level is better than bad Aikido training.
Sometimes what really needs to happen if you wnat to really do Aikido is that you have to move. Just a fact. To paraphrase Ikeda Sensei, "it's not Aikido that doesn't work, it's the particular practitioner's Aikido that doesn't work".