I understand why you would think that but then I think you think the government has the actual power to completely proselytize society which isnt always the case because if it was I think there'd be a lot more students doing STEMM (apparently UK students kind of avoids maths etc).
Take for example my situation which will certainly be similar to many other situations and I'm guessing you might be in the same one too.
I enjoy economics, I could do law as I currently take it, I'm pretty good at it and I think it's interesting. The gov could 1) increase the capacity of law places at uni 2) increase pay lawyers get (however this depends on the companies who demand lawyers etc) 3) subsidise my training until I'm a lawyer (or firms could). however it still wouldn't be enough to convince me to take such a path because money for me, like a lot of people, isn't a key consideration. Job security is more important but people still take subjects like: philosophy, pharmacy (supply > demand), sociology, psychology, business (S > D), economics, ("
, History, RE, Certain Languages, Law etc. though you could argue all subjects have such limitations, those are just the ones that come to mind.
And even then I'm not that concerned about job security, It's not like I'm guaranteed a job, I care more about research, learning and discussing what I like with others.
You're right that some people may be persuaded to switch jobs but those jobs that the government may want to increase the supply of will most likely be: Judge, MP, Surgeon etc. Roles that take skill and qualifications (1, you've already disincentivised a group of people), a lot of time (2, again some people won't be willing), hours can be long and stressful (3 I think you've just knocked away 80% of the people who were interested) and the most important variable: will it interest you for 40 or so years or can you really say the pay is worth it?
Essentially even if the governnment tried to campaign for more people to do engineering at a professional level, or let's say computer science the only people you'd convince are those who were already interested or leaning towards it dubiously not those who don't want to. I can't imagine someone trying to do medicine for the sake of 75k a year if they originally wanted to do literally anything else, they would've had to have had some pre-existing interest and the types of roles the government may want you to go in as I said before will be roles where there's a shortage in QUALITY; the genuine reason why the gov may act. There's no way they'd offer teachers 100k a year because you are right it would divert labour from certain areas.
However under the branch of let's say surgeon some people from different branches may switch due to money as an incentive, say from heart to brain; although this isn't a great example a lot of people would still do medicine if they could.
I think it's a good idea if they can control it. I don't actually know how much doctors earn but the gov needs to influence and disincentivise effectively to ensure that we have the correct amount of labour but they can't or at least refuse to due to issues with money. Control is important because we don't want wages to fall rapidly due to oversupply but we also don't want people to have to wait 7 hours so see a doctor.
And considering money is probably the only long term objective, aside from getting a job, let's assume someone did indeed switch degrees to do a much harder course (training too) in the hopes of one day earning a lot of money, disregarding all of the factors that will weed a person out let's assume he gets this job as an engineer, as a banker, as a barrister. For one thing the person has the skills and qualifaction and so quality won't be affected, in fact it might improve regardless of his motivation; in fact because his motivation is his/ her money it may result in even higher output as well as better quality output and society would gain a lot from this. Perhaps they may not be as happy as they would be if they did what they'd always wanted to do, but it seems to me like this would meet the compensation principle even if it is at the sake of the pareto criterion
Basically if this is to have any major effect they'll have to start them off really young so primary schools. Even then people have preferences. If there are shortages in let's say social care workers, demand will increase for them and thus price and thus supply once people become aware and the cycle continues making the government intervention essentially unnecessary.
Though that's just an extreme model aha