As others note, it is hard to negotiate gardening leave.
As others haven't noted, gardening leave is not to stop you from doing malicious damage. That's what firing you is for.
Gardening leave is primarily a competitive process, to stop the loss of company resources. Most commonly, in banking or management consulting, these resources take the form of relationships. Danny, working at CompanyX, might have an excellent relationship with the CEO of a company. When Danny announces her move to a new company, it is very plausible that the CEO of CompanyX will want to continue working with Danny.
To mitigate this, CompanyX mandates that Danny must go on paid leave for 3 months. The aim here is that the CEO still needs work done during these 3 months, so Danny's replacement begins to foster a relationship with the CEO, and progress is made on whatever the CEO needs done. When Danny finally returns from her trip around Asia, enough has changed and enough new work has been executed that it is too costly for the CEO to follow Danny.
As such, the investment the company made in Danny taking 3 months off was substantially repaid by stopping the loss of the client, whose billable hours are, of course, worth far in excess of Danny's salary.
As to your question - it is not sufficient to have a customer facing role. You need to have a customer facing role with a senior manager - eg C-suite - for a role where you provide most of the value, eg Management consulting or banking.
If you are customer facing and you're supporting a product your company sells, then even if you move the product is still there. For example, let's say I'm Atlassian, I'm hiring you, and you're the Atlassian rep at support Ford Motors. If you leave, even if you work with the ghost of Gerald Ford (who is somehow the CEO), it's unlikely that they'll drop Atlassian to pick up Fog Creek software. The value is the product, you're incidental to the relationship.
On the other hand, if you're advising Gerald on the structure of layoffs, or where to build the next plant? Most of the value is with you. On the other hand, if you're not working with Gerald, but only Jimmy Carter (who is only mid-level ghost management) then even though he might love you he cannot sign off on taking up whatever company you've moved to - so again, it will be hard to negotiate gardening leave.
If your company offers gardening leave, then it's because you're likely capable of taking a client with you when you move. If you cannot do this, you can negotiate gardening leave if you offer to take one month's pay instead of, say, 2 months (ie your notice period is 2 months. You offer to leave the next day for one month's pay, your company saves one month's pay).
If you are a manager (the more senior the better), you could negotiate gardening leave if you, say, agreed not to hire employees from the current company for some period. I've seen that done before - all that happened was people were poached a little later.
In Your Particular Case
Being customer-facing, as noted, isn't enough. It needs to be with key decision-making personnel where you are the product. Leaving to different industry and a non-competitor further hurts your chances - it's unlikely you could reliably use your direct reports, so offering a non-poach agreement wouldn't be useful (as you probably couldn't poach anyway).
Your best bet would be to ask for a reduced total leave period, and for that reduced period to be as gardening leave. Perhaps even just 2 days / week?