Here are some thought about the important question frequently raised - and we are not brave enough in to go into details. Large organisation are facing with the issue of complex "hairball" architecture, which is not performing well. Frequently the changes, which entitled to fix the performance results more issues and even worse performance. Performance: this case it could mean all the business aspects, including reposnse times, TCO, time-to-market and so, finally the aspect of Ability to change.
The most important question you would raise about any issue you analyse by "fix or replace" reason: would I become good to great? Fixing existing applications or the whole architecture usually drives you from bad to good - but never to great! The reason is declared perfectly by the baseline of quality management: if you do all time the same thing you will still get the same result. If you recall the history of problematic components you will remember how many times you started them to fix; how many hope-cycles were started which tend to be the boring delusion.
On the other hand, replacement has no warranty that the result will be great. It's a promise, a fantastic one, but you have have to turn it to a great result and not an other delusion.
Money
If you will start a replacement or a fix of a set of solutions, you will prepare BC for them - but it is not realistic to keep up-to-date BCs for both large scenarios to decide where to go. Although Cynefin framework is far to be the best to evaluate complexity it has great attributes collected to realise where you are and what would be the best direction for you. Cynefin and other things about complexity is written in Everything you always wanted to know about Complexity article, reading that you will learn a lot about identifying complex situations and measuring complexity. The most important part for the current topic is the table summarising Cynefin. If you are in Chaotic or Complex, or even worth the Centre of Disorder (when you cannot ercognise where you are, or you feel you are in all the places in parallel) I suggest you to aim "tabula rasa", otherwise Fix is your valid way to go.
There is one more thing on the money aspect. Assuming the case that your company is going to the market situation when revenue declines while the owners still want to get better profit ratio. It is simple mathmatics: you have to have cost level decline with better gradient! If this case happens worthwhile to forget about fixing, you need a new world, means the only way is replacement. The other question is if replacement must be application replacement or vendor model change is enough... Vendor model change means outsourcing or vendor consolidation usually results application replacement at the end, since the vendors are not from Roxfort and they do not have thier magic wand.
People
Raise the question frankly: do you have heroes or you have repairmen? Having heroes for fixing an architecture will result a real chaos, since they will introduce fantastic new "bridges" to overhaul the legacy problems; introduce new world with mechanics will have the result of rebuilding the same boxes now from Lego and not from wood-bricks.
This aspect is a real management challenge. I would refer here the Fiedler contingency model, which discusses the importance of the relation between management style and organisation needs. Just substitute the "manager" woth you team and the "organisation" with the solution needed and you will see useful lessions reading the contingency theory.
I do not now that kind of scale for the architecture LPC (Least Preferred Co-worker) of Fiedler, but again: I suggest to use that simple pairs Replace-Heroes, Fix-Repairmen.
I would discuss one more important thing here about people! Assuming you can decide which direction fits you by the People aspect is ideal. On the other hand if other drivers are selecting the way you go, the team must fit for that also! If you have to replace because of any reason be aware to get some superheros onboarded, while the fix case you have to keep focus on mechanics instead.
Company Culture
On the first sense this is very the same issue as People. A culture, which is not open for change will never be successful in a replacement; the opposing case is hard to imagine, since companies with "hero"culture are continuously changing the things therefore they do not reach the point of fix needed.
Still their is a very important topic to touch: starting to change people parallel the change of culture. I know that you company is doing well but theoretically there is a mistake, that management starts communicating the change, employing people fits to the communicated new way while forgetting to really change the culture making it ready to reach the target.
Ways to fix
There is the certain way of fixing: analyse, understand, recognise, fix. No questions but a well-known always repeated engineers' practice. The only problem is that you have to analyse, understand, recognise and fix. Is there any other way? I believe so. The way what I used many times dips a bit from the "tabula rasa" case sometimes but it is not necessary.
Let me tell you a small tail about human beings. All of us are lazy. Someone is more others are less but most of the people is doing the things to reach good enough results. But never to be perfect. Fixing requires more than good enough, therefore you have to wake up people. The magic word is gaming, which means a kind of competittion. Remembering to my father who said he would die if tried run around the block, but he was happy to run to catch a ball for an hour - someone call it football. You have to hunt for opportunities to initiate competitive situation around topics to be fixed, leading people to run to catch the ball, and let thing becoming better.
If I say competition here I do not suggest to build a full blown parallel solution next to the old one, because that is too much. You need to run a 1.5 player or by other words a challenger game. You should find a challenger with the potential to change the game, but with conditions which makes the game cool and hard enough. It is a sensitive balance but you are a great architect to do that.
If you do not care to keep the competition situation crispy then the forces pushes the competition are being liquidated and going to sleep. This is another reason to avoid having same size competitors because their situation used to result a non-competitive balance by cutting the cake between themselves and the only thing you will have some great professional set of communication about competition. The show part, not the results you were fighting for.
Ways to replace
I would focus on the team aspect about replacements and not the technology side. A simple reason is that with a good team you will find the good solution also. There is a fantastic book discussing a lot about this question, James Collins: Good to great.
In fact, leaders of companies that go from good to great start not with “where” but with “who.” They start by getting the right people on the bus, the wrong people off the bus, and the right people in the right seats. And they stick with that discipline—first the people, then the direction—no matter how dire the circumstances.
Jim Collins: Good to Great
Referring again to the football. 2018 chanmpionship knows that team is more than the collection of the best ever players. You have to fill all the seats in the bus.
You have to build or find the right team what will really deliver your target. A replacement program always has high risks, therefore a team is crucial for the success. One or two heroes will die on the barricades; repairmen are not the risk takers.
While change is risky you may want to analyse more, but please avoid to wait too much about replacement! Pick the example that the root case of change is a technology becoming deprecated on the market or inside your company. More waiting never helps.
A good advise for replacements:
Never try to implement Your Past 2.0!
You will get a new solution trying to simulate everything you had but will be never similar, therefore it will get negative judgements only!
Baseline
I left it to the end since it is so important for both cases, the up-to-date knowledge about the architecture. The article of Building a live architecture knowledge base describe you have to be up-to-date, but there is a learning for us here. While you build up your knowledge base you may have the feeling that there are too much secrets or indisspensable people. If they are be brave and start replacement immediatelly, this case is a barrier of fix!