Lol, it's noteworthy that Bonafacius had pitifully little strength to deal with the Vandal invasion. The African force mentioned by the Notitia Dignitatum apparently existed only on paper. The rest of the WRE was little better off, so they had to get the ERE to help, with Aspar, not that he was able to do much....The point is, it would appear that even before the loss of North Africa, with its resources, the WRE just couldn't raise adequate forces from its citizens (nor, after 408, would most barbarians serve in its regular forces). Recovering Africa would've helped but it would still be treating the symptom not the disease.
That is due to many factors. For one, Aetius left North Africa out to dry.
Second, North Africa was, before that point not under any serious threat, or at least, apparent threat like say, the ever troublesome Gaul and so on the list of priorities for military assets, it was fairly close to the bottom.
Third, one of the reasons the West was so chronically short of cash and manpower was to a certain extent a demographic problem, as Latifundia run by major landowners were not willing to let their tenants join the army rather than earn them money but those same Latifundia were increasingly reluctant to pay any taxes at all, and what taxes they
did pay were often being appropriated by corrupt officials before they ever reached the treasury. Add to that the resulting ridiculous tax burden placed on the middle class and suddenly they are cutting the feet out from underneath the economy. It wasn't that the funds did not exist at all, it was that all the money in the empire was rapidly being collected and then sat upon by an increasingly small group of people. Money doesn't do the economy any good if it isn't moving around.
Once North Africa was lost, whatever remained of the regular Roman army ceased to exist. They were forced to turn now entirely to feodorati and other mercenaries that already had equipment and training and just had to be paid. They still had some bite, as Flavius Aetius proved when he beat the Mongols, albeit with a lot of allied aid. However the age of the Professional Roman Army in the West was over, as Africa, which was second only to Egypt in wealth as a province, was no longer available to provide food and taxes to support the army. (This, in a unrelated note, made the Senators and their vast estates ridiculously powerful, as suddenly it was only at those landowners' will that taxes and recruits could be found, along with food for the cities, such as Rome.)
The point is, you are right in that it wouldn't solve every problem, nor allow Majorian to suddenly raise new legions and go forth to throw back the barbarian hordes, but if he managed the conquest well and made sure to keep the powerful landowners from Italy out, he would have a vital economic engine back. This source of taxes (and more importantly grain) would not only supply food for the cities of Italy, but reduce the power of the landed senators, get some cash flowing into the treasury so that Majorian can start recruiting and training native troops again to at least reduce, if not immediately eliminate, the reliance on feodorati and if he is smart, he can use that strength and those more reliable troops to begin the process of breaking up the power and estates of the Senators. I don't know if he would be successful and like so many emperors, all the good he might do could be undone by his successor, but hell if he survives an attempted assassination he has all the justification he needs to seize the plotters' estates and begin parceling them out to smaller freeholders. Maybe go back to the old mainstay of rewarding veteran soldiers with land to help secure their loyalty and help rebuild the Empire's manpower pool.
Taking North Africa makes it a lot more possible to start doing all of those things.