It's difficult to tell what you're really asking here. It's sort of like asking if stocks will be a legitimate asset. The short answer is: some will, some won't.
It's not meaningful to look at all cryptos together as "legitimate assets" because they're all different and will eventually succeed or fail based on their own merits. During the last bull run, a lot of projects did well and went up in value simply because the whole market was rising. Since the market was rising, prices in BTC were going up, so people kept buying because they were "speculating" that they could make money. This happened even for tokens that had no projects behind them; they were just a token.
It would be more meaningful or relevant to ask: "Will any cryptos gain mainstream acceptance and/or adoption?"
The answer to that is different, and is probably what you're looking to know.
The short answer is, "yes." Whether we like it or not, blockchain technology is here to stay, and some of the cryptocurrencies that are built on them will be adopted. It may take more time, but it will happen.
The control that it affords the issuers of centralized cryptocurrenies is too much to ignore. Having a currency that is blockchain-based would give a government complete control over where those funds go or don't go. It would be as simple as blocking withdrawals from a specific wallet. This isn't the dream envisioned by most crypto enthusiasts as they see decentralized crypto as the end-goal. It will likely be usurped by governments seeking more control over their citizens.
It might be a longer answer than what you were looking for, but hopefully it helps you better understand that not all cryptos are the same and shouldn't be judged together. Many sub-par products and projects have fallen apart during the current bear run. Because the funding isn't there, they have been exposed and have collapsed. True, some legitimate projects have closed as well, but it's a natural part of business cycles. The projects that have a competitive advantage are more likely to succeed, while those that are just trying to cash in on the crypto craze will fail.
Be careful and make sure to do your own research before buying any crypto products.
No. Simply because I think that there will always be nations who try to ban them.