On a very broad/general level, it's not really very different from a solid-state (SS) design: the solid-state design likely uses orders of magnitude more transistors! Op-amps use a minimum of 10 or so, and modern highly refined designs likely number in the low 100s.
The difference of course, is cost as you say. The cost of a transistor is asymptotically approaching zero. The cost of tubes was always fairly modest, and is only likely to rise in the future. There are other reasons why tube op-amps were rarely used (they perform relatively poorly), but broadly similar techniques could indeed be applied at audio frequencies, if we didn't care about tube count.
The easiest way to increase gain of a tube stage, is to increase its voltage gain dramatically. We can increase output resistance by using a cascode triode configuration, or a tetrode/pentode; and we can increase load resistance by using a gyrator (see "mu follower") or constant current source (some designers permit semiconductors for somewhat "passive" purposes like this; LND150 and similar types are a common choice). This replicates the VAS (voltage amplifier stage) + CCS load that is typical in SS amplifiers (discrete power amps, and internally in many op-amps). A pentode + CCS can have a typical gain over 60dB, which can then be reduced to a controlled level by negative feedback, along with its distortion, and output impedance.
Depending on permissible distortion, and available phase margin, active filter techniques can also be applied. Perhaps instead of RC sections to ground, they could be applied between input and output (as feedback). Active filters, using tubes, tend not to perform well, again due to their limited gain, but if just a modest boost/cut is required (say +/- 20dB somewhere), this might be an effective solution.
In the given schematic however, many stages are required because of the labeled purpose -- overdrive (OD). To generate significant distortion, plate output signal levels of some hundreds of volts are required (that is, a sizable fraction of B+, which is nearly 400V here). (Notice the cathode follower with "99% attenuator" to get a reasonable line level to external effects (FX)!) From even a high-level guitar or external preamp, this will require a couple stages of 12AX7. Which is a cheap enough way to get there.
Perhaps the cascade of triodes could be replaced by a single a pentode-CCS circuit as above, but you're burning at least two sections anyway; and since combined triode-pentodes exist, you're still using one whole tube to do it. In contrast, triodes generally have lower noise than pentodes; 12AX7 is also a very common type (still in production, no less), making this an effective all-around solution.
Even more importantly, and a more subtle point -- the cascaded stages will have a different sound (distortion spectrum / IMD vs. signal level) compared to a single stage. There are also time-dependent recovery effects in play here (the coupling capacitors are charged by over drive + grid current), which may exhibit a different time constant than the guitar's own decay response, which can potentially give a surprisingly nuanced envelope (attack-sustain-decay) response; with the precise distortion spectrum changing throughout the envelope. Offhand, a single stage will prioritize more even-order harmonics (mostly just the top, or just the bottom, peaks of the wave get clipped), whereas the cascaded triodes could be tuned to saturate at similar levels (top and bottom get clipped at similar levels); and a pentode+CCS type will saturate more sharply/suddenly than a triode, giving a harsher sound (which might play better say for heavy metal than jazz?). The quality of the distortion further affects the harmonization; a softer or cleaner sound will make better use of many strings simultaneously, while even just two strings might sound quite dirty or disharmonious when played through heavy distortion. (Which, as far as I know, is one explanation for the preference of, say, 3- and 4-note chords in clean-guitar genres, versus power chords (root and 5th) in say rock and metal. Or for that matter, tritones, when an even more disharmonious sound is desired, as often the case in heavy metal.)
