I think we need to be clear what we're talking about here. This is being presented as father's rights, whereas to date it has been the child's right to relationship with both parents. Where are the equivalent mother's rights?
Presently the mother has all the responsibility. This may appear as having the 'upper hand' as far as the father is concerned, but legally that's not the case at all.
If we're talking about parental rights, rather than child welfare, where is the mother's right to be able to rely on the other parent taking equal responsibility?
Paying child maintenance is the one that immediately springs to mind, but as a working single parent, I know that the responsibility and the cost of arranging childcare around a job was 100% my responsibility, while my X has the benefit of childcare on tap with no costs whatsoever. I'm the one who takes off time for sick children, doctor's appointment, etc. My career path has been permanently altered and continues to be restricted, whereas his is unfettered. There is no acknowledgement of this - in fact single parents are told, in the main, to 'suck it up'. They are disparaged daily in the national press, the government takes little account of the childcare difficulties faced them and has cut back financial support for it. And at the same time single parents are labelled with thinly veiled disguise as responsible for the moral breakdown of society.
Coming back to maintenance, 60% of single parents who do not use the CSA receive nothing. Half of those using the CSA pay £5 or less. We treat parking fines more seriously than non-payment of maintenance in this country. Charging for the CSA will mean even more single parents taking 100% of the burden financially since it's a 'get out of jail' card for many non-resident parents. This is being done at the exact same time the government is positing plans to give greater rights to fathers. 
THere should be no plans to increase father's rights until equal measures are taken to ensure maintenance is paid, since the numbers not getting maintenance are far outweighed by the number of father's desperately trying to get contact with their children and being denied.
But coming back to child contact. What's going on here exactly? How many mothers denying contact are doing it with 'no good reason'?
I've met loads of women whose partners have taken them down the legal route for contact - then failed to take it up once they've won it. Why are they doing that? It's because it's about control rather than a desire to genuinely spend time with the child. THe mothers know this, which is why they fight it. To the outside world and the professionals, it appears as though the woman is obstructive, but the truth is she's trying to protect her child from being used as a pawn in a game.
Domestic violence is much more prevalent among single parents - it features as a reason for relationship breakdown in about a third of cases. Yet because the vast majority of it is unreported and therefore unprovable, it is discounted. Since DV is a feature in 75% of child abuse cases, and research shows a significantly increased risk of a child being abused if one of the parents is abusive, I'd say DV is a damn good reason to prevent contact or at least insist on it being restricted or supervised. But if it can't be proven, the resident parent is accused of lying, making 'spurious claims' or being 'intractably hostile'.
Some women stop contact because contact was being abused by the father - letting down and disappointing the child and used as a weapon to mess up the resident parent's life. Some men who claim their Xs won't allow contact, really mean : I want to be able to see my children, whenever I want to, with no notice, to pop into the mother's house because I'm passing, to change plans if I get a better offer or even fail to turn up completely if something else comes up and I forget that an extremely excited child is waiting on me.
When you have a child, your priorities change. If they don't, something is a bit amiss. I don't think anywhere near enough significance is given to this. One of the reasons mothers feel that the father's involvement is undesirable is not because they no longer love the father and therefore want him to conveniently disappear. It's because they recognise that juvenile, irresponsible behaviour is no longer appropriate when a child is around. Often it is the father's refusal to accept that and to change that results in the relationship breaking down in the first place. Most mothers love their children passionately and may be preventing/restricting contact to protect their children, not to 'spite' their Xs. I'm not saying that's right, but I think a greater understanding of this dynamic would help and possibly avoid protracted court battles.
Resident parents can be reassured that poor parenting doesn't necessarily mean damaging or abusive and is worth putting up with because of the positive benefits associated with the child maintaining a relationship with the other parent. However, in the main, a very clear message need to be sent to those wanting contact - if you feel the need to take your child to the pub/play 18-rated games/drop them off with the grandparents/be hungover/sit in front of the telly/bring your new girlfriend of 4 weeks along/not turn up for contact/alter it without goodreason - on the exclusive day you have the opportunity to spend time with your child, then clearly you have not understood what contact is about - the child's right to a relationship with the non-resident parent. Those sorts of behaviours do not develop a relationship and so I'd question whether it's worth continuing. I'd like to see contact withdrawn when parents abuse it. I've seen too many children devastated by being let down by daddy yet again. Research conducted in the last few years has demonstrated that unreliable and uninvolved contact is almost as damaging as no contact at all.
This is a gendered issue because 92% of single parents are women. However, male single parents face very similar problems to the female ones when it comes to contact and maintenance, which shows that this behaviour doesn't happen because of biology. This isn' about hating men but addressing a cultural imbalance that devalues caring for a child, which is mainly done by women.
I don't deny that some women deny contact out of spite. Women are not saints and they share the same spectrum of behaviour as men. However, in the vast majority of cases, I believe women have a good reason for denying contact, even though it may not be a correct one (e.g. worries about the standard of care provided by the NRP).
So, by all means enshrine the father's right to contact in law. But ONLY if maintenance is made compulsory for all and properly punished if it isn't paid, cut off contact for NRPs who mess about with it, and introduce automatic supervised contact only for parents with a history of DV.