You are mixing up important things and probably that's why you don't see a bigger threat here than the livelihood of a couple of lawyers.
People who attended law school/university/have a law degree are not necessarily practising law as lawyers/attorney or whatever they call them in the given country, as a member of the government. But of course, once a marine, always a marine...
On the other hand, the parent was trying to refer to criminal defense (Amendment 6 of US Constitution). That's not something active politicians can participate in (conflict of interest, etc.).
If one has the means to pay for their criminal defense (or family law, anything that truly matters for the average citizen), they tend to be willing to pay a lot and expect some quality of service in exchange. They will certainly not choose ChatGPT, just because that provides the answer faster for long documents.
But a large number of people don't have this kind of money. As long as governments don't have a proper budget for legal aid, they will be inclined to cover the vanishing legal aid budget with false claims that they fulfill "appointment of counsel" with LLM services - while they don't really care about the outcome of the case. (And of course, successful politicians have rarely worked as pro bono counsels before their current job, not that it has any more relevance here.)
Non-Western governments are also interested in having this replacement service - the cost of lawyers is also money there. It's much better for all governments to have chatbots they have reliably trained than less controllable humans.
That's a hypothetical threat for now, but I hope you understand why there is something more here beyond "guildthink" and protecting the livelihood of lawyers at all costs behind the Sixth Amendment. There are lots of countries (or even within the US, like in Utah) where you can review contracts without being a practising lawyer, like LPOs mentioned in the paper. So far, it failed to make the world a better place.
But replacing including LLMs as "counsel" under Sixth Amendment would mean a very different situation.