They get infinite appeals rather than one appeal then deported on rejection, which means they stay here indefinitely, and have lawyers from the profesional working class despising folk of Starmer's ilk that help these migrants.
They don't get infinite appeals (nobody does in any variety of Tribunal/Court within the UK).
If the Home Office makes a decision which an applicant disagrees with you can apply for an appeal to the First-tier Tribunal (Immigration and Asylum Chamber) on basically any ground that you fancy (though if you would like to win it is wise to advance relevant ones that show how the Home Office decision failed to apply the law correctly).
If you lose at the First-tier Tribunal you can apply to the Upper Tribunal (Immigration and Asylum Chamber). But you can only appeal to the UT if the FTT's decision was legally flawed. It is no longer sufficient that you disagree with the Home Office decision, you have to show that the FTT decision was legally flawed. For instance it did not apply the correct law, or wrongly interpreted the law, or there was a procedural irregularity, etc etc. So before you can have a substantive appeal to the UT you have to apply for permission to the FTT, if they refuse you can apply to the UT directly for permission to appeal. If the UT refuses permission that's basically it. Your appeals are over the decision stands. If permission is granted then you can actually proceed to appeal to the UT.
If you win at the UT you might be remitted back to the FTT for them to have another go at getting it right or alternatively the UT might make the final decision there and then (depends on the actual issue with the initial appeal and the information available to the UT judge).
If you lose then you can appeal to the Court of Appeal but, again, only if you can show that the UT/FTT made a material error in the application of the law or a procedural issue and there is a realistic prospect of success. That you don't like the decision is insufficient grounds to appeal. You'd apply first to the UT and if they refused, you can apply directly to the CoA. If the CoA refuse then that's it. Your appeal is dead the decision stands.
If you are granted permission but then lose at the CoA stage then, you can appeal even further to the Supreme Court but same hurdles apply. Permission must be granted and will only be granted if there's an error in the law not just because you don't like the decision. If permission is not granted the appeal is dead, the decision stands.
Theoretically if the Supreme Court refuse you could probably go off to the European Court of Human Rights but the number of cases that are going to be granted permission to appeal to ECtHR is going to be vanishingly small so in reality the buck stops at the Supreme Court.
Now, you can argue that there are too many steps but I would note that firstly, there are not infinite appeals. There is a clear appeal system with specific steps along the way. But more importantly after your appeal to the FTT you'll only get further appeals (to the UT and above) if you can show that the decision made was wrong in law. Doesn't matter if you think the decision was dumb and wrong. Plenty of dumb and wrong decisions are made perfectly legally everyday. And that applies at each stage of the process up the hierarchy.
So the appeal system acts as a filter preventing meritless appeals from occurring. Even if every asylum seeker appeals their Home Office asylum decision to the FTT (which I don't believe is the case), only a fraction of those that then lose at FTT will be able to appeal further. Most that lose will lose at that point and their appeal will end, the Home Office can then remove them (and if they don't that's a Home Office issue, not an issue of the Tribunal system). Plus even a win at the UT stage doesn't necessarily mean a win overall. Perfectly possible to win at UT, have a fresh appeal in front of a new FTT and for that FTT make a correct negative decision.
Sadly, of course, that system is tremendously poorly understood (and it seems that those who are most keen to whip up ferver against asylum seekers are least likely to be interested in explaining it) as with so much of the Court and Tribunal system so people get it into their heads that all those rejected asylum claimants are swanning around appealing willy nilly indefinitely delaying their departure. Of course the real reason so many of their departures are delayed after a loss is due to the rank incompetence of the Home Office rather than repeated appeals.
Anyway, apologies for the wall of text but I get frustrated when I see easily disprovable information being bandied about regarding the Court and Tribunal system (of whatever flavour).