Along with this system, I would like it if basic jobs could still advance, and those advancements would be used in cross-class skills.
I say this, because while the current cross-class system is... underwheming, it's a good idea and opens up a lot of different potentials in terms of builds and playstyles. My fear is that once specialization happens, the basics go away.
Basics in this case being the basic jobs. I personally always feel it's a waste to throw away stuff, but I'm not against re-doing it if absolutely necessary. With that said, should basic jobs advance along with specializations and advanced jobs, you could get a deep system that allows for plenty of customization.
Being able to switch specializations without great penalty (or none) I think is needed. As FF Online's hook (in a sense) is being able to experience everything without starting over.
In the end, specialization skills can only be used on that spec, while basic skills (such as ones you get from PUG, CON, etc) can be used on all jobs regardless of spec. Currently the motive to level jobs high is to be more versatile, and I like that the progress I've made on one job could help the job I want to play.
Another thing that worries me generally about this is SE's ability to design useful skills. Of course, every job has filler skills and useless trash, but in XIV I see a lot more useless skills and WS than useful ones. This is likely a side-effect of cross-class, but the skills seen are quite lacking in imagination and even more in utility.
Edit: As for one of the systems suggested last page, I think that a specialization should be accessible from certain jobs, but having two different types of WHM depending on what class you choose to get there is an unncessary step. Leave that stuff for the specialization tree to handle. I like the idea conceptually, that a WHM that came up from a THM is different than a WHM that came up from a CON, but that's a lot of fiddly design that's somewhat unnecessary.