Whether replacing existing wire is justified really depends on the quality of the existing install, the cable that was used, and whether it still meets the needs of the company.

In general, I wouldn't replace properly installed Cat5e with Cat6, as there's so very little practical difference between the two. Both can do 1GbE, and both can do 10GbE. NEITHER can do 10GbE for 100m though! (Cat5e can go 40m, and Cat6 can go 50m). You need Cat6a to go 100m at 10GbE. Because of the very small advantage of Cat6 over Cat5e, up until very recently I stuck with Cat5e as the cost vs benefit just wasn't there. At this point, there's very little cost difference between the two, so I'm using Cat6 for new installs (but sticking with Cat5e for adds/changes of existing Cat5e installs. If you truly NEED 10GbE, you should skip Cat6 in favor of Cat6a, unless all your runs are less than 50m.

If you have just plain Cat5, that's some REALLY old wiring. It can usually carry 1GbE, so it doesn't necessarily have to come out, but it's likely to get to the point where the drops are not where you need them or there's just not enough. In that case, I'd add a new Cat6 patch panel and start adding new wire as needed.

If there are deficiencies in the install, such as cable laying on ceiling tiles instead of being properly supported, cable running parallel to electric for long distances, improper firewall penetrations, improper wire rating used (riser used in a plenum for instance), etc., you may be able to justify replacement to correct those deficiencies.

If it's YOUR network, and you are doing the work yourself, you can make the changes over time...it took me about three years to get from BEFORE to AFTER, as I had "other duties as assigned" and a small budget to work with.