Why hasn't there been a serious effort to develop an anti-inflammatory that blocks the responsible genes?
Your question is very complex, so for now I'll focus on a few points and answer these first:


  Why hasn't there been visible progress on a gene-blocking approach? There are all sorts of ways to block them - find all the proteins and other products that promote or cause inflammation, then consider all sorts of options to stop them. Options including blocking transcription, blocking translation, stopping the protein/product from exiting its cell, blocking its receptor, blocking intermediates.


You are right that all of these options to block genes exist for laboratory or research purposes. Using them on humans as medicine is a lot more complex in a lot of ways though. One main factor I'm not going to discuss much is the cost of all these techniques - which generally a lot more compared to making a simple chemical drug.

Blocking translation
This is a almost a standard experiment for research labs and can usually be done quite specifically (without side effect on most other genes). In oder to achieve a transient (short lasing) effect you usually use RNA knockdown: The introduction of small RNAs, that specifically target the mRNA of the gene you want to knock down, into a given cell with stop transcription in that cells. There are two problems with using this in patients: 1) transformation efficiency: you need to bring RNA into to the cells, which is not a very efficient process. You'll need even higher doses of the stuff to get a strong enough effect in the immune cells you want to target, since it's very hard to specifically target them. 2) The transformation process itself requires either potentially toxic chemicals or viruses to get the RNA into cells, both of which a costly to make and no one wants to take them unless they have to. Taken together this makes RNA knockdown a very bad drug / treatment strategy.

Blocking transcription
The only way to block transcription of a gene permanently is gene editing - which you really don't want to as a short term treatment against inflammation. To make it work transiently you'd have to bring proteins into the cell that bind to and block the genes you want to suppress - inactive transcription factors would do the trick. One (almost minor) problem is that they wouldn't be very specific, so you'd get lots of side effects. The bigger problem is that it's very hard the get a lot of proteins into cells. You'd need large scale transformation and there are many problems with that, which I discussed above.

Stopping the protein/product from exiting its cell
This is an interesting idea, which might actually work - given a few additional years of research. It's probably not possible to this specifically, so you'd generally block the release of proteins from cells. However if you find a drug that only lasts for a few hours it might work. One important caveat is, that the same (or at least a very similar) mechanism is used, when synapses release neurotransmitters - which you definitely do not want to mess with.

There seems to be active research in this area

Blocking its receptor / blocking intermediates

This strategy is very viable in the immune system, where many soluble factors like cytokines with respective receptors are important. Blocking of receptors or soluble proteins can be done by antibodies and there are many people working on this, some antibodies are already licensed by the FDA as drugs for humans. In most cases these are however intended for use in cancer or other more severe diseases, as all of these are immensely expensive.


  Future "last resort" treatment for chronic debilitating inflammatory diseases could include "nonsense mutation" gene edits that permanently disable inflammation in specific cell types.


You do not want to this. Inflammation is an important process that keeps your body healthy and permanently disabling it would basically make you immune deficient, which is not a good thing.


  The cost might be prohibitive with today's technology although it would almost certainly become feasible with biochemical 3D printing.


Biochemical 3D printing will likely never be a thing. It's possible to print e.g. organs with cells, but printing proteins requires molecular construction and it's unlikely we're ever ging to find a system that's better than what nature came up with during a few million years of evolution.
In general though technological advancements should decrease the cost of all these strategies.