What is to prevent the rich from bullying the poor is the rule that if they lose (which with an unmeritorious lawsuit is likely) they will have to pay the costs of the defendants' expensive solicitors, who took on the case despite their clients' poverty because they knew that a successful defence would be paid for by the losing plaintiff. This is precisely the reason that the rules were changed in the 1990s, to allow conditional fee arrangements.
Unfortunately another problem rapidly appeared; poor applicants can demand money from rich people, knowing that going to court will cost the defendant more, both financially and in publicity, than paying up; even if the plaintiff is ordered to pay costs, he cannot pay what he has not got. You don't need a good case for this, just an unscrupulous solicitor; and shockingly it appears that there were some such around. There is no perfect solution, but one useful weapon was reviving the ancient rules against maintenance, by which 'investing' in a lawsuit in order to take a profitable cut of the damages was banned, as likely to clog up the justice system with speculative cases. The Lord Chancellor's Department (now the Ministry of Justice) has tried hard since then to balance the two problems (and simultaneously to reduce the legal aid bill for political reasons); how successful they have been depends on who you ask.
And any judge in any country has the power to add a party to the case if it would benefit justice; whether adding lawyers purely so that they can be responsible for the costs is such a benefit depends on the system. Without such a rule (and of course the power to direct unreasonable costs to be paid by the person who caused them win or lose, which exists even in most American states though it may be rarely used), there would be nothing to prevent both rich plaintiffs bullying poor defendants and poor plaintiffs blackmailing rich defendants.