Most modern websites use some sort of 'wildcard' search, however they use it without special tags. Instead of specifying a special operator (like *) for a wildcard search they have a smart algorithm.
For instance, when I type in 'alaxy s 9' in the search bar on amazon.com I still get hits for the Samsung Galaxy S9.
The system matches my search query 'alaxy s 9' with the Galaxy S9. Which is probably what I mean with that search.
Advanced searches, like the wildcard operator and others like 'OR' / 'AND' are becoming more and more outdated. People do not use advanced search anymore. So if your team members mean these operators, yes they are becoming out dated. However, it is not hard to still support them. Likewise offering an 'Exact match' option can help users that know what they are looking for.
The goal of a search option is that people can find what they are looking for faster. For instance, have you noticed the trend of auto suggestions in searches? Nearly every big E-Commerce website offers this tool.
The auto suggestion is some sort of modern 'wildcard' search.
The 'wildcard' search method in itself is not good or bad UX. The results you show make it good or bad UX. If your search algorithm is not capable of showing the correct results users will become frustrated and that is bad UX.
For me personally I expect a website that offers a search function to show me what I am looking for. With the example above of Amazon, when I type 'alaxy s 9' I expect to see the Galaxy S9.
If you wish to do some further reading on search results you can read a recent article of Kate Morgan from Nngroup. It offers some other great insights on how you can provide better results for your users.