Summary: the impact of "influencing policymakers" through advocacy is often vastly over-estimated; working as a civil servant is likely far more impactful. (This view is specific to UK political system and UK civil service. Views are as a result of decades of personal experience across multiple government departments, working with ministers of every political persuasion).
I feel my heart drop a little when extremely passionate and well meaning people tell me their aim is to influence policymakers.
I almost don't want to tell them that MPs don't write laws. Secretaries of State aren't swayed by that one thing someone said at a roundtable. Ministers take decisions, but almost invariably (95%+) agree with the recommendation of their civil servants.
A civil servant will always be trying to ensure that their recommendation is aligned with the government's agenda and the broad steers and ambitions of the minister. But the majority of the prioritisation, the policy proposals, the approaches and the detail are all the work civil servants. Ministerial speeches, Op Eds in newspapers, comments at a panel, are all written by civil servants. Civil servants recommend where ministers should go, who they should meet, which ideas, risks and concerns they are exposed to. Civil servants draft new laws. Civil servants write guidance. Civil servants steer how millions of pounds of money is spent. I estimate that civil servants are responsible for 90% of object-level decisions. 90% of those decisions are the maintenance of the status-quo.
However, there is vast scope for impact as a civil servant working on policy. In particular, in neglected and/or new and emerging policy areas, civil servants set the agenda and frame the terms of the debate. Furthermore, the level of influence and scope for impact is not directly correlated to seniority.
Decision makers, whether politicians or civil servants, are constrained by the systems they operate in. Even if you were to convince them that they should make some new legislation, it's highly unlikely they would be able to do so at that point in time.
Changing policy is not about who has the most robust arguments, or even the best evidence. It's almost entirely negotiating the art of the possible within a risk-averse and largely conservative (small c) system.
Effectively shaping policy usually looks like:
What research can you commission that's then published by government, which then becomes part of accepted government narrative?
Which leading campaigners can you build alliances with, who then become trusted sources of advice to ministers?
How can you make it easy - self-evident - for your seniors and ministers to say yes to a very reasonable, small step in the right direction?
How do you build on that and create momentum?
Which questions need answering?
Which questions already have answers, but need to be answered by a trusted authority in order to be accepted?
Is this the most effective way for this budget to deliver outcomes, or might there be an alternative approach?
How can doing the right thing deliver government and ministerial priorities?
If you are strategic, have strong analytical skills, good communication and influencing skills, then the expected value from becoming a civil servant, even if initially working on an area unrelated to your prioritised cause, is far greater than starting a career in campaigning or advocacy, particularly if you are aiming to influence politicians. Politicians and civil servants are largely immune to changing their opinions as a result of interaction with campaigners. However, I grant that as a campaigner, influencing public attitudes at scale may be more impactful than trying to influence politicians.

I would agree for Westminster, but this is relatively tractable in LMICs where governments rely a lot on external NGOs for policy development and implementation + US legislators always have the habit of adding riders to bills. The title may be misleading unless qualified.
Edit: As a former civil servant in Singapore, and as someone whose friends are all still serving, our favourite joke is that we signed up to do evidence-based policy-making, but what we often do instead is policy-based evidence-making (and this is in Singapore, which is famed for fairly extreme technocracy).
Anyone who joins a high-income country civil service to make a difference should expect being stalled by bureaucracy (plus legitimate constraints on policy options, whether political or budgetary or operational, that aren't necessarily apparent to the public). Also, it really makes a difference if it's a politician pushing it vs someone in the civil service hierarchy; civil servants are deferential to democratically elected politicians in a way they aren't to their own peers or juniors in the service. Suggesting new ideas is met, at baseline, by the dead-eye stare of someone who knows this just creates more work for them.
Thanks, I have amended title for clarity.
I think your final point is part of what I am getting at - that policy doesn't happen on the basis of good ideas alone, and there is bureaucracy, and there operational, political, and budgetary constraints on what can be done. As such, just suggesting good ideas or compelling arguments to politicians or civil servants isn't what works, whether as a civil servant or a campaigner. What works is providing neat approaches which work within the system, which use the political, budgetary and operational factors in their favour, and which make your peers look good, perhaps even giving them the chance to claim it as their idea.
There's huge friction within the system, and you can either see that friction as a barrier, or you can see the friction as the thing that gives you grip, the fixed points that you can use as leverage.
That's what I mean by policy being the art of the possible.
Whilst people here will be swayed by evidence and logical argument, we risk wasting resources if we act on the assumption that they are sufficient to make impact in the wider world.