Best Integrated All-tube Amp?

I do think you are being a little too picky in your requirements. I would suggest broadening your search and then audition some of them; you might be surprised of the results and how they might conflict with your prejudices.

I own an Audio Valve Assistent integrated. It's supposedly rated at 20 or 25 W/Ch, but I consider it to be more like 10 because it has a hard time driving anything except very efficient speakers. When mated to an efficient, tube-friendly speaker, though, it sounds excellent. But alas, it's push-pull, so it won't satisfy your stringent requirements.

I also have a Jolida 302b and for the price I cannot recomment an integrated more highly. I have a nice "big system" that sounds great, and adjacent to it sits my "little system" which is powered by the 302b. Many of my friends actually prefer the little system.

The Jolida can drive real-world speakers that the Audio Valve can't. For most people, it would get my recommendation. If, however, you have efficient, tube-friendly speakers, then the Audio Valve will astound you.

I think you're going to have to drop some of your requirements to find something in your price range; just let your ears be the guide rather than the specifications.

Michael

I own an Almarro 318, and use it in my video system driving a pair of Diatone single driver loudspeakers in bass reflex cabinets, all sourced from EIFL in Japan.
The amp is truly musical and full range, and very much underrated, using the same output tubes as the Lamm mono amps in Class A single ended mode. When I tried it listening to CD's in this configuration, the resulting detail and musically involving resolution were quite a shock.
I would happily use it in my main rig to drive my Ocellia Tilias, single Phy-Hp driver crossoverless speakers if my custom made 3D Audio 300B ever died on me - at least until I got it fixed! I don't foresee this happening, but I did use it until my 300B arrived, and its performance was right up there with the 47 Labs Gaincard it replaced. At it's current price its a steal.
I listen to everything from Ella to Marilyn Manson.

The Bel Canto SETi40 fills nearly all of your criteria. Uses 2 845 tubes and provides 40 WPC Pure Class A, zero negative feedback and and has a remote. Mark, at Bel Canto provides exceptional Customer Sevice and Bel Canto are an established firm.
Not cheap, but used, it is affordable. Read the reviews on Google.

Why not hybrids ??

I am not aware of an integrated amp that meets your requirements (single-ended, triode, 300B-based, class-A, no negative feedback, low distorsion, point-to-point wiring, good bass, and a well-known brand), especially one that is "affordable", whatever you mean by that.

The VAC Avatar is about the only thing that comes close, and I believe it is $3k-$4k used. The VAC Renaissance 70/70 Mk. III amp (not integrated) meets all of those requirements other than being single-ended (it is push-pull), but costs $5k-$6k used and you would need a preamp. The VAC amps do come close to the single-ended sound and avoid the significant disadvantages. Singer in Union Square carries VAC -- ask for Dave Lalin.