Willem don't take it bad, some amps depend on the preamp to give music a signature, my solid state Class A power amp is a buffer and its sound signature depends on the source and I won't feed such an amp with the MacMini DAC's signal nor I would bypass the AR SP9 and, sorry again, if you compare FirstWatt by Nelson Pass or KT88 tubes to a class D amp we are talking apples and clams
I've done a bit of digging and reading up on this system. It's a collection of vintage and less usual items that I'm not convinced about. But it is your starting point and you've chosen these components...
The first step is to get the speakers looked at, if you haven't already. The L220 is supposed to be good on the impedance front, and for the F4 to have a chance of working you'll need that to be right. If you have problem components in the speaker the whole thing is going to fall to bits, especially if the impedance falls in the bass, in which case the amp won't provide enough power. I think the F4 is borderline here in any case, but it's not the worst pairing you could have.
Newer, well designed speakers would make the biggest improvement usually, but the amps limit choice and you appear to be on a vintage kick, so if the L220s work and you like the sound, I'd say work with them. If you
don't like the sound, then lose the lot and start from scratch. You can build a good modern system for the resale price of your components and your apparent DAC budget.
Personally, I'd replace all of your electronics with a good all in one or integrated amp, and would suggest (since you seem to be a traditionalist) a decent Yamaha, or stretching to the Marantz Model 40 - class AB, more than capable of driving your speakers.
If you must keep the FirstWatt, I would strongly suggest you need a proper preamp. The SP9 adds distortion to the FirstWatt's distortion, the Beresford more again: and that would be too much for me personally. You may differ, of course. I'd be looking for a solid state preamp with low distortion and good voltage output and gain. Such things are expensive new, unfortunately. Benchmark (we get to be stuck records here, either Benchmark or Topping/SMSL) preamps would be a good choice. Two F4's run as mono power amps is another upgrade that may help with your speakers if you are committed to them. They are not recommended items here, of course!
A lot of the modern DAC/amps as suggested by some here will not work with the F4, which needs good voltage and some gain. And digging in, I agree the Beresford is the first thing that should go. A Benchmark DAC may be the best answer if you intend to keep everything else (again, we have to be a bit concerned about decent output levels from the DAC). If the Denafrips appeals, though, again it's not the worst choice. Any DAC recommended here will work with a proper preamp to the F4, and a cheap model gives you more room to work with the next steps. The Beresford isn't a great DAC, from what I can see. I suspect that it won't provide sufficient voltage for the F4, so wouldn't take the advice to connect it directly.
You do not need fancy cables and
especially not fancy AC cables. Even if you want them, they should come well down the list.
So, you are closer to correct with your ideas here than we may at first think. Forget cables, though: replace the Beresford with a low distortion DAC, and get the speakers properly checked and refurbished if necessary and doable, if you haven't done that already. Then consider amp choices. If the speakers have impedance problems such as a dip below even 6 ohms (I can't find a good measurement) then your system is probably mismatched, and you'll need to look at what you want to change.
