(disclaimer = not a solicitor myself so maybe totally incorrect in what I say)
It depends what you mean by 'judgement' - professional judgement or ethical/moral judgement? The former absolutely they should use, the latter not so much. A solicitor can certainly advise their client (fairly robustly) that an application is misjudged/unlikely to be successful and advise they go down a different path. That is very much their job - people often misunderstand the legal process or want to pursue fruitless applications out of spite or misguided belief it will get them what they want, and the solicitor should advise them of the likely outcome and the costs they will incur and try their best to steer them down a better path (although by 'better' that should be in their client's best interests and not necessarily those of any other involved party).
It's then up to the client (assuming the client has capacity of course) whether they pursue it or not, and unless there is something inherently unlawful or impossible to do what the client asks, the solicitor will usually follow through with making the application/writing the letter to the other party/whatever it is on their client's instructions, even if personally they wouldn't do that or think it's not wise - it isn't their role to make moral judgements on the value of the person's case so long as what they are asking for is within the scope of the law. So e.g. even a father who is obviously not safe to have direct contact with his children due to proven violence, drug use, abuse or whatever, might still properly have some very limited parental responsibilities/rights and it would be the duty of a solicitor to help him exercise those through the court if necessary, even if s/he finds the man morally repugnant personally. That's why people say the law is blind, because in a civilised society even the very least/worst members of our society should have access to their legal rights and someone to help them access those...
In this case there's nothing I'm aware of to stop the father trying to informally reach a parental agreement outside of court, indeed solicitors would usually advise this as a sensible first step to avoid the costs and hassle of going to court. Obviously if the mother of the child doesn't want him involved or doesn't acknowledge he's the father or whatever she won't necessarily agree to everything he wants or at all. She isn't under any obligation to agree or even to respond, just because a solicitor is involved. This is where then the court process and/or mediation would kick in, depending on the exact circumstances...