To answer some other aspects:
In the case of Starship, is the existing steel strong enough (to be
non-ablative!), or would a different type of material be necessary?
The melting point of most steel alloys seem to be right at the
In general, metals do not have good mechanical or corrosion properties when at the edge of the melting point, and the transition between cold and red-hot and back again will also often be "fun" -- after all, this is how metals are hardened, stress relieved, and annealed. According to the MakeItFrom materials database, various 300-series stainless steels have temperature limits due to corrosion or mechanical loss of strength between 400 deg C and 1000 deg C. Reentry temperatures seem to often be over 2000 deg C.
To be clear, there are plausible spacecraft designs where the structural outer hull is simply made of materials able to withstand the temperature of reentry, and the inside is insulated with appropriate insulation materials. These tend to be excessively expensive and hard-to-machine alloys involving nickel, molybdenum, tungsten, cobalt, etc. My impression is that the cancelled X-20 Dyna-Soar spaceplane was to use this strategy, with lots of exotic (for the 1960s) metals.
Can it be raised with paints and resins containing things
like aerogels or glass? Can steel with higher temperature tolerances
be made? - what are the limits here?)
You are describing an ablative heat shield. (many of these involve fiberglass and phenolic resin). Spray-on ablators for re-usable spacecraft have been studied.
Coatings able to reusably withstand high reentry temperatures will have to be ceramic, metallic, or metalloceramic. This sounds like the so-called "thermal barrier layers" used in jet engines.
Why not just fill the shell with aerogel? Has this been tried? To what
extent is the weight (or cost) of such concepts, prohibitive?
Aerogel is a very good insulator, but despite it being a ceramic-type material I do not see it usually proposed for extremely high temperature applications. It is extremely fragile as a bulk material and often therefore is used as powder or chips integrated into an insulation blanket. In any case, this would be a "hot structure" type of application as described above. Also, hot structure designs seem to have issues with weight (possibly because the high temperature alloys have poor strength-to-weight ratios and it is hard to build strong, light structures that can also survive being cooked).
Are there practical limitations besides weight, that might limit the
scale, such as heat dissipation, or thermal expansion? What techniques
have been tried to counter them?
Thermal expansion can make structure-building, especially using multiple materials, a lot harder. The Dyna-soar, for example, was to have its structural elements connected by hinges rather than normal rigid connections.
The heating profile over time also matters. Ballistic reentry tends to lead to very high heat for a short time, lifting reentry (like the Space Shuttle) leads to less heat flux but over a much longer time. "Heat sink" and heavy ablator heat shields seem to work well with the former option.