It's not a free pass going to higher pressure.
First off, any given pressure tank has a larger drawdown (tidal volume, amount of useful water stored) at 20/40 than any higher setting. Unless you increase the size of, or add additional pressure tanks, at a higher pressure your pump will stop and start more frequently (which is usually considered to be a large factor in how fast it wears out - running longer and starting less is better than running less each time, but starting more.) As you can see from the example chart below, for a given tank, 40/60 is already less than half the volume stored at 20/40.
This version is not quite as bad (they do vary a bit):
You will need to check your tank, but most pressure tanks are good to 100 or 125 PSI (the ones in the upper chart are only 75 though), so that's probably not the limiting factor. You'll need to check what the pressure on your hot water heater's temperature-pressure valve is, but it's probably also fine. My pump supplier suggested (and I installed) a 100PSI overpressure valve on the system as the pipe entered the house.
Many common items (toilet valves, etc) don't really like pressure over 80 PSI (yes, I know you only proposed 70.) 60/80 is about as high as you can expect to adjust most pump control valves to.
It's not the most common situation, but if your well pump is marginal for the depth you may reach a situation where it's cannot reach the higher setpoint as you use water and draw the well level down - so it will sit and spin but not reach the shutoff pressure.
If your system was designed for 20/40, I'd suggest doubling or tripling your tank capacity if going to 40/60, and tripling if going to 50/70. How important that is may be influenced by how deep and inconvenient and expensive your pump will be to change when its time comes. Cheap and easy, maybe you don't bother increasing the tank at all. Expensive and a major hassle - you might consider more tankage a good investment.
6 PSI amounts to 15 or 16 vertical feet rise, at most. 200 feet sideways and 15 feet up, not 200 feet up.
As such, I'd have to say your pump appears to be in need of service or replacement, based on what can be gleaned from what you've written. No water is going into the tank because the pump is evidently unable to make 30 lbs of pressure, much less 50. 6 PSI is not a functional well pump.
You described this as a shallow well pump - is it in fact a jet pump (recirculating, 2 pipes to the well), or is it in fact a straight shallow well pump (pure suction, one pipe to the well, less than 27 feet to the water surface)? A jet pump introduces a few more things that could be screwed up.
If you are facing replacement and it's a jet pump, I'd strongly suggest moving to submersible, as they are much more efficient than jets, (you could replace a 1 hp jet with a 1/2 or 3/4 submersible and pump more water in most cases) and with today's technology, also more reliable.
Best Answer
Unless you have a special type of modulating pump you will need a pressure tank. In most cases the pressure tank is used to stop the pump from short cycling. If you forgo a pressure tank the pump turn on and off continuously greatly reducing the life expectancy of the pump. Draining it would be recommended if you're not using it and away. If the place is cold you can also use RV Plumbing antifreeze to ensure no residual water freezes. Flush it out when ready to use.
Side note, After reading your other post. Yes you can operate the pump from the breaker with just a pipe connected to nothing. Be careful to not dead head the pump, meaning don't run it with a valve closed. Without a pressure switch to turn it off at high pressure you can hurt the pump or burst a pipe (unlikely but possible)