I don't disagree with the other answer but I think it would be helpful to add some points about UK law specifically, since that is tagged in the question.
For England and Wales, the role of the jury is subject to the Criminal Procedure Rules, which provide that anyone involved in a criminal case must (among other duties) deal justly and fairly, acquitting the innocent and convicting the guilty. They swear or affirm that they will give "a true verdict according to the evidence", which is to say that the verdict should not be based on such factors as the punishment the defendant might receive, or information not presented and argued in court. The position in Scotland and Northern Ireland is essentially the same.
The jury members do not have the right to step outside their assigned role as finders of fact, within the set process. The accused also does not have the right to tell them to do that. Rather, it is the accused who has the right to a fair trial, and the jury who have the shared duty to give them that trial. The accused does have the specific right for the jury to be impartial, and many other rights. The jurors are protected from reprisal for returning a not-guilty verdict (see Bushel's Case), but still have to follow the court's rules, just like everyone else in court.
It's true that the privacy of jury deliberations mean that they might not do their sworn duty, but as a matter of law they are meant to decide based on the evidence presented, and on the law as explained to them. Aside from contempt of court, there are now several specific offences for jurors who fail to do this - mainly directed at those who do their own research - but also including engaging in "prohibited conduct", which is
conduct from which it may reasonably be concluded that the person intends to try the issue otherwise than on the basis of the evidence presented in the proceedings on the issue.
This provision, introduced in 2015, is in response to a case of a juror who posted on Facebook:
Woooow I wasn't expecting to be in a jury Deciding a paedophile's fate, I've always wanted to Fuck up a paedophile & now I'm within the law!
Contempt proceedings followed (Attorney General v Davey  EWHC 2317 (Admin)) and the 2015 legislation was introduced to make that kind of situation easier to administer. In Parliamentary committee discussions it was suggested that this offence might also cover the giving of a perverse verdict, and although that intention was denied, the case still illustrates that juries can deviate in both ways from their duty. There are plenty of other cases about racially biased jurors, jurors who knew the accused or a victim, and so on, all of which are potentially compromising to the proper basis on which they should decide.
Barristers may be found in contempt if they suggest the jury should disregard the instructions of the court. The judge has the last word on jury instructions, literally, and is in a position to give them the proper basis on which they should decide, or in more extreme examples to discharge the jury and start again. So it is a bit tricky for a barrister, or a litigant in person, to sneak through the suggestion.
In response to your enumerated concerns -
- There are many appellate cases concerning judges' instructions to jurors, and the expectations in the summing-up are well established. This will include language explaining how the jurors are meant to proceed.
- It is not a question of absolute trust but a system of overlapping safeguards. These include early removal of potential jurors from the pool if they cannot promise to participate properly, the various briefings given to them, the possibility of being held in contempt of court, the possibility of inquiring into jury deliberations if there is misconduct, the jury policing themselves within the room, the eventual option of returning a majority verdict (e.g. 11/12) if there is a stubborn minority, and the fundamental asymmetry between prosecution and defence.
- It is not the case that litigants in person are able to suggest that the jury should not follow the law. There are lowered expectations from the court about how they might behave compared to professional lawyers, and they don't face the possibility of professional sanctions, but the judge is still entitled and expected to stop them from messing up the proceedings. More commonly, LiPs face other obstacles in properly presenting their case, and the current policy is that judges are meant to nudge them back into bounds without impairing their right to take part. So judges may have a different manner with LiPs but they are not exempt from the rules.
- There are some ECtHR cases about jury trials (and recall that not all European jurisdictions even have juries). The Convention language requires a fair hearing by an "independent and impartial tribunal", and there is a right for the accused to participate effectively. But in Miljevic v Croatia  and Brandstetter v Austria , it was held that this does not entail "an unlimited right to use any defence arguments", and in Melin v France  that he has to show diligence himself if he refuses legal assistance. In Legillon v France , it is restated that in a system where a lay jury returns a verdict without having to give reasons,
Article 6 requires an assessment of whether sufficient safeguards were in place to avoid any risk of arbitrariness and to enable the accused to understand the reasons for his conviction. Such procedural safeguards may include, for example, directions or guidance provided by the presiding judge to the jurors on the legal issues arising or the evidence adduced, and precise, unequivocal questions put to the jury by the judge, forming a framework on which the verdict is based or sufficiently offsetting the fact that no reasons are given for the jury's answers.
This would suggest that the regime in England and Wales is compatible with the Convention rights, given that its overriding objective (from the Criminal Procedure Rules) is to guarantee a fair trial. While "the rule governing the secrecy of jury deliberations is a crucial and legitimate feature of English trial law which serves to reinforce the jury's role as the ultimate arbiter of fact and to guarantee open and frank deliberations among jurors on the evidence which they have heard", Gregory v United Kingdom  ECHR 9, the potential for arbitrariness arises and that is offset by the surrounding procedural safeguards. Again, some of those exist for the protection of the accused against prejudiced jurors, or jurors who do outside research and so might find negative information that the accused had no chance to see and counter in court. The suggestion in the question about the jury being "ignorant of their full rights" does not apply, since the jury members do not have Convention rights (such as free expression) which would operate in this context, where they are required to be impartial in the interests of justice.