You can ask for deferral, but it is not universal that you can/will be granted it even once, and it is much rarer to be granted it again. With that said, I don't think it is a question at all of merely ego or vanity - humans universally have this annoying tendency of wanting to be able to obtain suitable food and shelter for themselves, along with other things that comprise a decent standard of modern living. It is not unreasonable of you to infer that if a group has a variety of funding options they could offer and they choose not to extend it to you, that this is a valid way to interpret that their priorities are elsewhere. You need not read into it as any deeper judgement - it is what it is, and any such ordering of priorities includes a large component of abitrary and inscrutable caprice.
If you require funding, you require funding, and that is that. I certainly insisted on it, and decided in advance that any school that did not offer funding was automatically off the list. It is reasonable for you to do likewise.
You are certainly free to ask about an option to defer to apply for funding again, and tell them you cannot accept an offer without funding, and they will do whatever they decide to do. As a rough likelihood assessment, I think it is most probable that you will need to apply elsewhere - to multiple schools you would like to attend, not just one. Even if they grant you a deferral they will almost certainly not include any guarantee of funding next round. That's just how graduate school is: no guarantees of admission, funding, success in the program, or success beyond. Life in an ivory-coated nutshell, if you will.