If you are new to amps or haven’t used tubes before, I would definitely recommend starting small, figuring out your preferences, and then building from there. One thing that most experienced people in this hobby will emphasize is “synergy”.
I’m an electrical engineer and I could spend all day explaining the different topologies for both solid states (e.g. discrete vs chip, Class A vs D, etc) and tube amps (e.g. OTL vs OTC vs SET, hybrids, etc). Those thing definitely matter when considering pairings, but ultimately, often times there are certain characteristics about a given amp that TENDS to pair well with some headphones over others. That’s what I mean by synergy. And these characteristics can range from drastic (solid state vs tube) to subtle (output impedance of 1 Ohm vs 10 Ohms) differences. If you are interested in the actual electronics of amps I have plenty of resources I could point you toward.
But let’s start with “more power” as you mentioned. Typically, people love to recommend “more power” in general because it is usually a good rule of thumb that more power is better for headphones. However, really, at some point more power isn’t going to add anything extra and becomes overkill. There are very few headphones that truly keep scaling with more and more power, the Hifiman HE-6 is one that does. That is why it is common for people to run that headphone off of speaker amps. But a headphones like ZMF Atticus (which I own) does like a good amount of power, but that isn’t nearly as important as what kind of amp is powering it. The reason Zach himself recommends Beyer A20 is because that amp has a higher than normal output impedance (OI) for a solid state amp. That amps has an OI of 100 Ohms compared to say Magni 3 which has an OI of 0.3 Ohms. Why does output impedance matter? Damping factor. There is a common 1/8th rule used that states a given amp should have an output impedance 1/8th that of the headphone impedance. Read that source for the reasons why. Atticus’ impedance is 300 Ohm. So technically, the A20 violates this rule. But from a synergy standpoint, higher output impedance amps bring a special flavor or characteristic to Atticus. This is similar as to why the 120 Ohm output impedance Bottlehead Crack is known to pair well with 300 Ohm HD650. I haven’t heard the A20 myself, but I trust Zach so I do think that could be a great solid state pairing. Another option you may want to consider is Cayin iHA-6. That amp usually goes for about $650 new, but can be had used in the $500 range. It also has a higher than normal output impedance and I believe Zach has also noted it is a good Atticus pairing. Grizzlybeast wrote a great review for it.
Now, the reason you hear people recommend tubes for Atticus, and I also heartily recommend tubes as well, is because Atticus’ character lends itself to playing well and benefiting from what tube amps can bring. Atticus is naturally a warm, thick, rich, and meaty sound. If not paired well with amps I find it can sometimes feel a little too overbearing and congested, and not open enough. It isn’t terrible by any means, but Atticus can scale well with different gear. So I personally like tube amp pairing that emphasize a fast, clean, and airy sound. That is why I use Valhalla 2 with Atticus. That synergy makes Atticus feel a bit more lively, spacious, and clear. But that is my preference and how I like to hear Atticus. You could be looking for something completely different. That is why if you don’t know what you like, you should just start small and give yourself time to understand what you do and don’t like. You can always scale up gear later once you figure that out.
For tube recommendations, I would probably say Valhalla 2, Massdrop CTH, and Lyr 3 are very good under $500 option. I have the Massdrop EC ZDT Jr. on order myself as Zach also liked that amp with Atticus, but I haven’t heard it myself. It might be replacing my Valhalla 2 once I get it.
I’m less familiar with solid state amps. If you need a starting point I think Magni 3 represents a great entry level value. I have no idea what headphones or other amps you currently have, so that could be important to consider too. And as @taronlissimore mentioned, factoring in a DAC is important too.