Wait, you're nickname is Future and you're looking at using D5 ;-) As to your question... and apologies for this long ramble...
I would not necessarily second guess their reasoning without more information.
I would also say it depends on what you mean by a redesign. If you're updating the look and feel, but not really changing the basic function of the site, and it's running in D5 and you're happy with it, then why spend on a D6?
If you're talking about seriously redoing the site, including adding a lot of new stuff (you say your site "will have" various sections, but don't say what it *does* have now), then I would certainly consider D6.
Your devs may be reticent to use beta-stage modules on a client site and that is certainly understandable. Many of these modules specifically warn that they are not for use on production sites. So your developers would have some liability there and, in their shoes, I might do the same.
It's hard to say, but I would guess these guys are one or more of the following:
1) just trying to save you money and hassle,
2) super conservative and risk averse,
3) have a substantial code library they haven't updated yet which they want to leverage
4) just haven't kept with the times.
How I felt about their advice and whether or not I took it would depend on how good their story was.
FWIW, the drupal.org site only made the upgrade to D6 in the last month or so. They had a lot of custom modules for handling the load and other things and just couldn't convert until now. So obviously some of the best drupal developers in the world thought that the flagship drupal site in the world couldn't go D6 until recently, but they *did* finally pull the trigger. To me, that says D6 is finally truly ready for the real world (I've been using it for production sites for many months, but nothing high profile like drupal.org). They did a major Code Sprint with a large number of the best developers in the world coding like fiends as a team for three days or so to pull off this "simple" upgrade. So it's not necessarily completely trivial.
Still, that's an *upgrade* of a site that's running and has millions and millions of hits per month. Building a new site in D5 at this point, seems kind of foolish to me. A major makeover of a D5 site is sort of a judgement call. If I didn't *need* something that's in D6, I wouldn't bother.
Drupal 6 *is* a major improvement, but I was faced with a similar situation when D5 came out. In the end, I skipped from 4.7 to 6 because I did a test install of D5 and *for me* there was little or nothing that D5 offered and some of my key modules never really got D5 versions until D6 was out and both module versions were being developed in parallel. So I waited. Based on that logic, you may or may not want to stick with D5.
D6, however, offered some things I really wanted and D4.7 stopped getting security fixes, so I bit the bullet and upgraded to D6. D6 just feels like an application framework that was built on purpose. D4.7 was kind of "ad hoc" and D5 halfway in between. But I would not call myself an expert on such things (don't be fooled by the moderator title - I'm definitely not a drupal guru).
As one example, a properly coded D6 module has full uninstall capabilities that return your DB to the state it was before, so you don't build crud in the DB, something that always bugged me about Drupal in the past (pretty sure that's new to 6).
And yes, some features are built into D6 that were modules in D5.
As for building with non-release versions of modules... I'm building a site now that uses all the bells and whistles - a pretty complex affair sort of like building my own Facebook (obviously missing a few things, but also adding things that FB doesn't have - and no, I am not trying to compete with FB! It's totally different, just giving an idea of functionality).
The point being, I'm using several "dev" modules (no release) or one in release alpha, beta or Release Candidate status. Aside from a couple of minor bugs, which are often present in the D5 versions too BTW, in general I have had few problems. A few odd bugs, but nothing serious. Meanwhile, I take full advantage of D6 and am better positioned for the D7 upgrade in a year or so.
If you are making any custom modules, the module file set, the install schema, Forms API and the Menu API all change substantially between D6 and D5. They should stay roughly the same between D6 and D7, so your upgrade path from D6 to D7 will likely be fairly smooth, whereas going from D4.7 to D5 to D6 (there is no direct path) was a bear for me. A lot of futzing with this and that and editing raw data to get everything to work out. Obviously, that will depend on the modules you're using. Some have nice upgrade paths and some don't. The killer for me was Gallery2 integration, which you probably don't have to deal with.
Finally, don't wait for D7. Drupal used to be on a "one release per year" schedule, with six months in development and six months in code freeze. There was talk of sticking to that but integrating unit testing so they could go nine months in dev and three months in testing.
The problem was that module developers, many of whom are just plumbers or accountants who create a module for their own site, were unable to keep up and it was all becoming a mess. I never understood it and, like many, complained about it often. I think the message got through, because D7 is not actually even in code freeze yet and the most recent info I saw (in the last month) said that it would not get a full release until late 2009. At that point, it will take many months for modules to updated. So I would not plan to upgrade to D7 on a production site until mid-2010.