That voltage test seems okay, but remember, there is no load on it, so heat related things would not show up possibly.
I like to think of a transformer in a simple form. Take a nail, and wrap 30 wraps of wire around one end, then get another wire and wrap 60 wraps of wire around the other end. These would be 1:2 winding ratio, however not nearly enough iron to do anything meaningful. The idea, however, is an AC signal into one winding, will create an ever changing magnetism in the nail. The nail would then create an AC signal out the other winding. The ratio of the windings tells the step up or step down of voltage. An output transformer is a step down, in voltage, thus an increase in current and resulting impedance output. The Impedance of an output AC signal would equate to it's ability to resist being changed by a given load.
You can see that the input and output windings must be electrically separate, two different pieces of wire. Any DC resistance less than infinity, between the two windings would indicate a breakdown of insulation between the two. Your showing of 700 ohms, once verified, would indicate failed insulation between the two windings. In audio transformers this is a somewhat common failure mode, because the two windings are interleaved, like a few layers of primary coil, then a layer or two of secondary, then a few more layers of primary, and a couple more of secondary. This is done to cause the transformer to pass signal at various frequencies. A power transformer doesn't have the same issues, because the AC is all at 50-60 hz. A single frequency is a piece of cake, and the nail transformer could even do a fair job at low power levels.
Because the only connection from input to output windings is a magnetic one, a steady DC voltage on the primary does not induce an output on the secondary. Our tubes, being a changing resistor in that circuit, causes the 300-400vdc in that winding to shake, and we have an AC signal of, lets say, 100 or so volts AC riding on top of a 300-400vdc voltage. Think of that 400v as that AC signal's ground reference, in other words, 400v is it's zero, because it is from there that the AC voltage climbs and falls a somewhat equal distance above and below. The transformer does two things. It strips off the DC voltage, and only the AC gets to the secondary, and second, it changes the AC voltage and current. That 100+ volt bounce in voltage might only produce 10vac at the output, but at a higher amperage than the input. An isolation transformer is one with 1:1 winding ratio, and, in a perfect world outputs the same AC that goes in. It's the same job a coupling cap does when it passes AC signal, but stops any DC from going through, which means that each winding could be at a different ground reference, like one at 300vdc and the other at 0vdc, such as the case of an output transformer.