What is likely a much larger concern is producing Pu-239.
A reactor burning pure U-233 would produce very little in the way of transuranics.
The thing about a liquid fuel reactor though is that since you don't need to disassemble it or even shut it down to add more fuel you have an unprecedented flexibility in terms of what you can burn. Within certain limits you can use almost any fuel mixture.
The most obvious way to scale up a MSR industry is to start out burning a traditional U-235 / U-238 fuel mix while you breed U-233 from thorium. Then once U-233 production is sufficient you can quit enriching to obtain U-235 and just replace it with U-233 while still using U-238 (natural unprocessed uranium) for the bulk of the fuel. If for example your MSR design can tolerate 10% U-233 (U-235) / 90% U-238 then you can scale up the industry ten times faster than if you tried to start out using 100% U-233 from the very beginning.
At some point in the future when you are no longer increasing the reactor count at a rate that exceeds the achievable breeding rate of U-233 you can start increasing the U-233 / U-238 ratio until eventually you no longer need U-238 at all.
Along the way you can decide how to allocate your spare neutron budget. When you no longer need to breed U-233 as rapidly as possible some of those spare neutrons can be used to dispose of problematic isotopes, decommissioned nuclear weapons, etc, but in the early stages you are running reactors that use a substantial amount of U-238 so they would produce as many transuranics as traditional reactors produce.