Yet our heroes often only barely survive their combat engagements. Building six weak ships in place of one strong vessel would mean wasting all the resources, as none of those other ships would be useful. Except in noncombat applications, perhaps, but those weren't part of the original argument.
True, but often in situations that represent an unlikely, unforeseen set of circumstances, which in the vast majority of instances would not have occurred. The series is written as entertainment and as such requires drama. For most ships, most of the time routine missions would in fact be exactly that and not require a battleship or the equivalent.
Conversely, Starfleet is clearly already starved of resources in peacetime, as it has too few ships to deal with border incursions, colonial calamities or key scientific observations, forcing our heroes to respond solo, after the nick of time.
Exactly, so the best approach might have been to increase the number of available hulls. Assuming that heavy firepower on a large ship is an expensive investment why have that investment deployed well away from potentially contentious areas on a purely scientific mission which more often than not would require little more than a Runabout or at most an Oberth.
We see no onscreen disadvantage from "dead" weight, "excess" onboard personnel, maneuverability, or shield characteristics. No vessel in Starfleet is more agile than the E-D, in terms of turn rate or the like. We have little idea of relative linear acceleration, but that only means there's no support for the idea of the E-D being sluggish, either.
Well, yes we do, the Defiant, the Intrepid, the Miranda, the Sovereign are all visibly portrayed as being much more maneuverable. It's hard to quantify just how much of a disadvantage that bulk incurs of course for the simple reason we have no idea how another ship would have fared in those circumstances. If only we had software that could run "what if" scenarios, pitting various ships against the same aggressor and playing a variety of tactical opt.....oh yes, star trek online, ahem.
Why not pack superior scientific abilities onboard a superior combatant when there is no known penalty other than, just possibly, at the construction phase? Obviously to be able to produce more ships - but is that really a factor? The science abilities have to go somewhere anyway. And anything differing from an 1:1 ratio sounds unsound: a science-heavy ship would just succumb to enemy fire.
Granted that the scientific equipment represents an unknown in terms of space and power usage, in fact it may well carry tactical advantages in terms of sensor counter measures, fire control, increased shield penetration due to enhanced analytical capabilities, etc.
However, probes require storage facilities, as do non combatant shuttles. Making best use of those facilities requires living facilities for whole teams of mission specialists, along with onboard scientific teams, lab technicians, and technical experts. All of which bulks up the vessel and shifts design attention away from tactical concerns. This becomes less of an issue if those facilities are on board a vessel which is never intended to see combat and thus requiring little beyond basic armaments. You might arm a scientific vessel irl to prevent falling easy prey to pirates, but you wouldn't equip her with ICBMs. Likewise a warship will have sufficient sonar and EWAC to deal with stealthed threats, but is unlikely to carry a radio telescope rated for astrophysics (although yes I am aware that some naval vessels do in fact carry some pretty advanced microwave and radio detection equipment - the point still stands that they would not be as extensively equipped as a dedicated ground based facility and the equipment they have would largely be primarily tactical in application)
That scientific equipment has to go somewhere, so why not on a science ship? All SF ships are multi mission granted, but typically to help solve the problems of limited numbers, with the "only ship in the quadrant" so to speak taking up the slack until something more appropriate takes over. Trying to shoehorn multiple specialist ships into one hull may not in fact solve that problem at all, effectively tying specialist resources up on inappropriate missions for which they were not intended but some other part of the ship was.
Also the issue is not just scientific equipment. The Galaxy is designed as capable of carrying large number of colonists, requiring mass living spaces and storage facilities, along with clearly being designed for comfort unseen in any other starship design.
As for volume to shield ratio, granted there are unknowns in terms of the technology involved but as a point of basic principle, assuming some spheroid surface to a shield then a given amount of energy emitted over a smaller surface area surely would indicate a greater field intensity, hence greater protection, else a shuttlecrafts shield emitters and power supply would serve equally well for a starship. By that logic given the same power supply and emitters in each case the smaller the vessel the greater the degree of protection afforded.
It's not hard to see the advantages of splitting tactical tonnage between multiple vessels, greater flexibility, the ability to police/patrol multiple arenas, to respond to multiple threats simultaneously without being hamstrung by diversionary tactics and as you point out yourself the fact that multiple ship engagements seem to favour the more numerous party.
