Well, that is a tactic, expensive on hulls but it is an option. A continuous emitter is going to be an invitation for the other side to assist in saving on fleet maintenance costs by scratching the skimmer from the fleet. The Russians have AEW capability, sort of, from within their own national boundaries that would seem to be without the risk of loss of hundreds of conscripted crew per target, and the ire from the families about being lied to by the Kremlin.
I really think that the whole Russian fleet should be emitting, it helps the Ukrainians, who could use more help from the Russian Navy much as they have been assisted by the Russian army strategic planning and logistics system.
The Moskva ought to have been able to handle the scenario, at least up until impact. Providing air defence for a task force over a large area is exactly what she was designed for. From the very limited information we have the observed (poor) performance seems most odd. I suspect there is a lot of the story that has yet to come out.
Suggesting that the Moskva should not have been emitting is like suggesting that any & all of the other S300 batteries should not be emitting in case they too become targets. That's not the way Integrated Air Defence Systems (IADS) work. Some things have to be emitters and take their chances. I don't think the Russians have done anything with their AEW assets on the south and east sides, so far I have only seen reports of them on the north side, often operating out of Belarus.
For Ukraine this is a very good result.