That being said, I fly one of those shiny RJ's and I can't see the engines on the airplane. I would level off at 1000' AGL, accelerate to VT, clean the airplane up, then begin a climb, and call for the QRH for an engine fire. I would ask for vectors back to the departure airport, if the weather is below CAT 1 minimums I would ask ATC to find me somewhere with good weather and point us in that direction while we work on putting out the fire. I don't care if it's just an indication, if the fire light is on I would hit the fire push and blow the bottle(s). The airplane will climb on one engine and they make airport specific engine out procedures for these situations. I wouldn't have any problem defending my decision if I blew a bottle into a perfectly good engine if I thought we were on fire.
Whether you can see the engine or not is only one part of the equation. "Visual Indications" are only one part of the 5-part FEVER test. I don't know if I'd shut down an engine based ONLY on a fire light with NO other indications of any kind. Does your checklist specifically say to? To me, creaing a self-made compounding emergency (fire light-only, and now added single engine) isn't that prudent. Why create an ACTUAL emergency (single-engine situation) out of a POSSIBLE emergency (fire light-only).
As I've mentioned before, a fire on an jet aircraft with pod-mounted engines isn't that critical of an emergency......as compared to a fighter with internal engines to where if you have a fire, your whole aircraft is on fire. So long as you can close the firewall shutoffs, it'll only burn out there. You obviously won't keep it running, but if you've shut it down but can't get the fire out, worst case it'll depart the airframe.
I think there is a need for immediate action items, but have a bazillion seems unnecessary. In a crew environment there's no need to start grabbing levers and pushing buttons before you assess the situation, this can cause bad things to happen. We're not single pilot and don't have bad guys on our tail, so our workload is considerably lower. Just as you train specifically for single pilot military/91/135 ops we train to operate effectively using 2 crewmembers.
What assessment have you done in your above example? Reading the bolded portion in your first paragraph that I highlighted, you assessed nothing......you simply reacted to a fire light that had no other supporting indications ("I don't care if it's just an indication"). Your non-assessment and onlty following the light indication contradicts what you wrote in the second paragraph and I bolded. And this had nothing to do with single-pilot or two crew. This is where airmanship and judgement need to be balanced with checklist discipline, and vice versa.
In the EP sections for all AF jets, there's a preface that says (paraphrasing) "....checklists are not replacements for good judgement and do not cover all situations. Perform only those steps necessary to take care of the situation at hand...." That preface is where the whole balance of airmanship vs checklist usage comes into play. We can't be checklist robots anymore than we can be cavalier about EPs. There has to be a balance.
Not bashing what you wrote, but it does create some good discussion points. That said, I agree with the other points you make. Reasonable and prudent.