I agree with fay. That's how it should start. Prime the carbs, full choke and bingo.
Check out your chokes. Put on full choke and then see if there is any more choke lever movement on the actual carbs. If there is still movement then you are not getting full choke when you pull the lever. It is easy to locate the choke levers on the carbs by just tracing the choke cable up from the bulkhead.
It is quite complicated to check float level on these solex's but quite easy to lower the level by putting a thin washer under the float needle if you think that may help.
I think your problem is simple and my bet is the choke mechanism is badly adjusted or the electric pump is set at too high a pressure. I think about 3 psi is what you need. Did you fit a pressure regulator when the pump was fitted?
I do not have an electric pump so if the car has not been started for a few days it does take a lot of cranking to fill the carbs, but when primed it starts beautifully.
When did this poor starting start? Was it by any chance after the 123 ignition was fitted? If so the timing could be too advanced, that would give difficult cold starting. One has to be careful when timing the 123 to get the static timing setting correct, the instructions, IMO, are not very clear.
keep us posted