VISA has some of the biggest dedicated data centers engaged in all those rapid transactions over a centralized network. While the thought of having LN takeover Visa is tempting, it is really difficult for a network of voluntary, bootstrapped nodes with several bottlenecks to surpass the processing capabilities of dedicated hardware like the ones that companies like VISA employ.

LN by itself should be quite good for those of us who want to use BTC. While its a nice thought experiment, it isn't quite possible for even LN to change the status quo on this one.

Lightning Network is Bitcoin. For Bitcoin to challenge Visa, a lot of other factors need to be resolved too, not just the scalability. Price volatility is a problem, as unstable money is impractical for business. Bitcoin is not fully recognized by governments and their stance could change at any time. Most importantly, majority of people does not see benefits in using Bitcoin as money - banks work quite well and smooth for most people, on average their user experience is far better than Bitcoin's. Bitcoin's main goal is to let people control their money, but this is not what most people are interested in.

The main issue is price volatility. This could be attributed towards the lack of regulation and low liquidity on the crypto market (compared to the traditional stock market). Volatility is linked to the price of Bitcoin relative to Fiat. But what if people use Bitcoin as a unit of account? Then, 1 Bitcoin would be equal to 1 Bitcoin no matter what. If we start to value things based solely on Bitcoin, volatility would no longer be an issue. People would just pay any amount in Bitcoin regardless of its current price on the market. Yet, this hasn't happened as the world moves around Fiat these days.

As far as the Lightning Network is concerned, I think it has all of the capabilities of surpassing VISA in terms of TPS. What's limiting the LN to go full speed ahead is the lack of mainstream adoption. Not to mention, the LN is limited to Bitcoin's own transaction capacity. After all, LN channels are opened/closed on the Bitcoin blockchain itself. With a limited block size of 1MB, the LN will not be able to expand towards new heights. The constant opening/closing of LN channels on the Bitcoin blockchain will only make matters worse in the long run because of the block size limitation. Who knows what will happen with Bitcoin in the long run as it becomes more popular in the mainstream world?

Nonetheless, the LN is nothing more than an experiment. Besides years of research and development, the off-chain scaling solution is still in its infancy. It won't be able to overcome VISA if progress remains slow. At least, people have a choice to skip high fees and slow transaction confirmation times on the Bitcoin blockchain. As I've said many times before, people could either choose the Lightning Network or an altcoin to enjoy fast transaction confirmation times and low fees. Time will tell us whenever the LN will be able to challenge VISA as a payment processor or not. Just my thoughts

So which one is it, it has the capability or it is limited 
That block limit will most likely stay like that for years, I don't see anyone ever challenging it, and not a single of the current dev proposing this so it will all have to come from the capacity of the channels and finding a way to keep the coins spinning around in the network, which brings us the problem with the limited service offered, and so on and on.

But in the end, why does it have to compete on Visa's terms?
I don't think that we will have to soon the number of customers using bitcoin to rival Visa even with the capacity, what Visa has and bitcoin will never have with a 3rd party service is the easy usage for their cards. You swipe your card, payment in seconds, card accepted everywhere, you lose your card you have somebody to call, you have the funds in your linked bank accounts insured, those are things bitcoin will never be able to do.
And a lot if not the decisive majority will appreciate those more than being their won bank like a lot of them don't like this responsibility that comes with it.

Before bitcoin can challenge Visa there are a lot of things that need to change, with one of the most important being how people think and act, and this will take a while, maybe even a decade is an optimistic scenario.

I'm kinda under doubts tbh. Bitcoin was amazing back when only few were using it. Many coins seem great before they get congested, and then you find out they aren't as strong as they were supposed and presumed to be. I honestly think the LN is no different. At one point, having enough adoption, the mempool will be challenged even with LN widely adopted.

As I mentioned in a recent thread of mine, the problem is with the first layer and not with the second one. Bitcoin will need some changes before it can technically challenge centralized fiat convenience. But let's not forget that technical specs don't always mean similar results.

If the Lightning Network becomes:
1.As user-friendly as VISA.
2.Capable of handling as much transactions as VISA
Then yes,there's a small possibility for LN to challenge the domination of VISA.
There's another huge requirement-Bitcoin must become as mainstream as fiat money(which is less likely to happen).

What's the point of Paypal,VISA and Mastercard running LN nodes?Will they get any major benefit from this?

Even now, if I accept high transaction fees on bitcoin network, I can get fast transaction. I don't have to wait for 2 business days and wait till Tuesday to get a completion for my transaction if I broadcast it in Friday.

With Lightning network and its future, I don't see why Bitcoin can not beat VISA but VISA will still exist and plays its roles as one of most popular payment gateways for people.

With irreversible bitcoin transactions, I won't have to worry about chargebacks. VISA's chargebacks.

Blindly saying that it will replace almost any conventional payment system is not a good thing for this discussion. I don't know much about Lightning Network or VISA or how they process their transaction but what I can say is that if LN wants to topple VISA out of its throne, they have to match the infrastructure that VISA already has and provide a concrete evidence that they are faster than anything else. Maybe you are right that LN will replace other payment system, I sure do hope so, cutting the in-between for transactions is a good thing after all.

I think it is a big....NO from me.

People find it difficult to manage the most basic Crypto currency concepts like wallets now, so just imagine what learning curve it will require from them to setup a Lightning Network configuration.  

The developers have a lot of work to make a system like this so user-friendly that a uneducated person from a 3rd world country, would be able to use this. ( I work on contract in Africa for some of my work and I can tell you that the majority of them, will not be able to use the Lightning Network as it is now)