meleagertoo is essentially right that a 737 crew should be able to fly a 737, no matter what the mode and what the situation. My guess - purely a guess - is that probably the F/O was flying and was given an extensive (18 second) briefing about what to do, then unfortunately cocked it up. We've all been there.
As you say, a proper crew should have been able to fly a go-around either manually or with help from the automatics, such as they are. Why this crew - or one of the crew - couldn't do so is the real question, and the answer will be to do with their training and what competency standards were required to pass the SIM in that airline.
'Children of the magenta' often gets trotted out, but unlike that crusty old guy, pilots of today must be able to programme and operate the automatics in today's aircraft and today's skies. There is no shame or weakness associated with using the automatics, and there are plenty of valid reasons for using them. It is not very satisfactory in my opinion to have to keep taking manual control when trying to fly on automatics which are design limited.
There is or used to be a driving licence only for cars with an automatic gearbox, and a person could obtain such a licence and drive only autos. I am not saying that we should ever allow pilots who can
only fly with the automatics, but pilots today do need to be able to fly
with the automatics.
In a situation like this, you are right, the basic-ness of the aircraft is not directly relevant, but we are naturally going to ponder if its design was a factor in any incident. The 737 is a 1950's electro-mechanical and hydro-mechanical aircraft with some electronics bolted on here and there. Literally - I was amazed to see a box labelled 'fuel quantity computer' bolted to the ceiling above the F/Os head in a
737-300/400 - why not put it in the avionics bay?
Although not directly relevant, we are bound to ask ourselves why this manufacturer has not updated their aircraft. Why does it still have auto-pilots and auto-thrust that can only do part of the job? Having to have one pilot looking inside to adjust the take-off thrust as you roar down the runway, is not very satisfactory - both pairs of eyes should be looking out at what is going on. I understand it would cost a lot of money to update and also about the commonality required by SouthWest, but the so-called MAX should have been a proper modern update of the 737, with much more capable and properly integrated electronics.
The pitch-power couple is not a particular issue by itself, but having to push and trim
forwards to go up during a go-around is quite a design flaw. If it can be designed out with modern electronics, then why not do so? If by some quirk of physics our cars all veered to the right with increasing speed and to the left with decreasing speed, we would all be used to compensating and would no doubt have a manual trim system to compensate. But it is so much nicer that we don't have to do this, and like-wise it is so much nicer not to have to constantly pitch trim in a FBW aircraft. We could all drive around in cars with non-synchromesh gearboxes, having to double de-clutch for every gear change, but it is so much nicer that we don't.
Drivers are able to concentrate more on the driving, and the driving task is much less tiring with modern devices. Ditto aircraft.