Bouncing through the links in the article you provided, it seems that the UK government felt that the EU's proposal to use its standard model of reciprocal arrangements of visa free short stays and listed paid activities that are exempt from visa requirements, would not allow it to claim it had control over its own borders.
Having refused to include a chapter on mobility in the Brexit agreement, the UK government rejected the EU’s standard offer of visa-free short stays – working up to 90 days in a 180-day period – and a list of paid activity exemptions that could exclude musicians, artists, sportspeople and journalists from the requirement to seek visas to work in individual member states. This offer was incompatible with the Conservative manifesto commitment to taking back control of Britain’s borders, the government claimed.
Equally the EU rejected the UK's offer of 30 days visa free work for musicians on the basis that that offer is already available to the whole world and offers the EU no benefit in return for its own concessions.
The EU declined the UK’s offer of 30 days’ visa-free work for EU musicians on the grounds that this is the UK’s existing standard policy and offered no added value to its members, and because it offered significantly less than the EU proposal at the mobility negotiation.
As appears to be normal in UK EU relations, both sides have accused the other of misrepresenting the situation.