> bitcoin takes ages to confirm

Ten minutes, and if you want to assume just a little bit of risk, you can simply wait until the transaction appears on the blockchain, which is pretty instantaneous. In all my bitcoin transactions, I haven't encountered a single double-spend.

> supports somewhere around 3 [three] transactions per second

This is flat-out wrong. Please check your facts before spreading misinformation. It CURRENTLY supports 7 [seven] transactions per second, and this cap was fixed in software, by limiting the block size to 1MB. It can be raised quite a bit (by raising the maximum block size), the devs were simply a little unsure what would happen, so they capped it at 7TPS.
Source: https://en.bitcoin.it/wiki/Scalability

> and shifts risk from sellers to buyers (e.g. compare refund possibilities with credit cards)

Yes, let's compare. You can always reverse a credit card charge, or dispute it. There are known scams done by abusing PayPal, for example.

> various hacks that are being proposed to increase the number of transactions per second

There are no "hacks". It would simply be: Increase the maximum block size. In all likelihood, things would still work fine. Please read the Scalability doc I linked above.

> hacks on a proof-of-concept

Do you realize this "proof of concept" has been operating for 6 years now? With only a couple of hiccups which were quickly corrected? And that there are almost (if not more than) 500 (!) other cryptocurrencies out there, based on the same idea or variations thereof?

> whether it's possible to have a cryptocurrency that doesn't rely on e.g. blockchain technology.

Unknown, unless someone thinks of one. Just like it was unknown whether a decentralized transaction ledger could be secured, before Satoshi Nakamoto thought of one. And when someone does, that could simply be applied to any existing cryptocurrency.

> without a blockchain or equivalent is beyond me.

There is something called "Proof of Stake". You can google. It has some possible flaws, though. There are also other hybrid solutions.

> the idea of being able to cryptographically sign over a digital $5-equivalent -note to someone else, instantaneously, and without relying on a third party

That is impossible. You simply can't "rewrite" the dollar monetary system to use a decentralized technology. They use entirely different semantics. Say I wanted to "convert" an actual $5 to a "new, cryptographically secured, decentralized" dollar. What authority (see, you can't have any authorities in a decentralized scheme) would validate the $5, and then what authority (there's that word again) would issue you a credit in this new scheme? The new scheme would inevitably have to be centralized by fallible humans, unless you came up with a robot that would somehow validate a $20 bill, then completely destroy it while issuing you a "crypto dollar credit", which would somehow magically stay pinned to the US dollar... See, it doesn't work that way. Even "real" currencies don't work that way, they can all be speculated on, it's just that their pool of currency is so much deeper and their penetration so much wider (and their value pinned to enough real-world goods) that nobody really notices the ripple effects day-to-day.

> Sadly, a lot of bitcoin's popularity seems to come from people who sincerely hope

Are you seriously trying to disparage early-adopters, here. The same early adopters who adopt almost all new things and push them out into the mainstream...

> it's probably a good idea to be aware of and filter out their voices

Yes. Screw those rotten kids excited about new things! Let's wake them from their dreams and poop in their cereals, and give them a dose of reality!