Elliott wrestled board control from top shareholder Vivendi this month after a two-month campaign to shake-up the way the French media group has been running TIM.
Beyond a governance overhaul, Elliott has proposed a spin-off and partial sale of the soon-to-be-created network company, a conversion of savings shares, a return to dividends and asset sales.
But it recently added it would be up to the new board and management to decide whether and when to consider such actions.
"It's clear (the board support) is there and I feel very comfortable in moving forward with what needs to be done," Genish told analysts in a post-results call. "I'm here for the long run."
Genish has made his staying on as CEO conditional on being able to execute the three-year plan launched in March focusing on a digital transformation of TIM, fixing its finances and getting back investment grade credit rating.
The well-respected telecoms veteran said any additional opportunities would be evaluated as they arose.
He reiterated towers unit INWIT was strategic for the group, especially ahead of the arrival of 5G technology.
He also ruled out any sale of its business in Brazil, adding consolidation there should be "assessed carefully" so as not to jeopardise recovery and cash generation.
"Everyone is convinced that TIM Brasil on a standalone basis is a lucrative asset that's doing extremely well," he said.
TIM has launched a process to put its fixed line network into a legally separate company. The former state phone monopoly will keep an open mind regarding a possible floatation of NetCo, but will insist on TIM retaining control of the asset, he said.
CONFLICT OF INTEREST
Genish dismissed any suggestion of a conflict of interest over Italian state lender CDP being a significant shareholder at TIM while also owning 50 percent of broadband rival Open Fiber. "CDP is a financial investor," he said.
The executive is not concerned about Italy's two anti-establishment parties set to form the next government, even if at least one of them has called for greater state role in the telecoms sector.
He said he expected any party to support policies that would favour investments in Italy's digital roll-out.
TIM reported better than expected 1.7 percent growth in first-quarter domestic sales, lifted by solid mobile operations. Analysts had forecast a growth of 0.4 percent.
The domestic sales rise was "boosted by a surprisingly strong wireless unit. It was the fourth growing quarter in a row after eight years of decline," Morgan Stanley said in a note.
TIM said comparable earnings before interest, tax, depreciation and amortisation (EBITDA) fell 4.9 percent to 1.89 billion euros (1.7 billion pounds), hit by provisions made for fines Italy imposed on the phone group as part of the so-called golden power decree which the group is appealing.
Excluding the non-recurring items, underlying core earnings were in line with market consensus.
TIM said Vivendi was no longer a party exercising direction and coordination over the phone group, which started after the French investor appointed two-thirds of TIM's board and named its own CEO as executive chairman.
Following Elliott's proxy fight, the focus shifts back to TIM's operational challenges, including its 25.5 billion euros of net debt and new rivals appearing in broadband and mobile.
A key headache is the pending arrival in Italy of French rival Iliad, which is likely to pressure margins.
TIM shares rose after the results but later retreated. The stock closed down 0.7 percent.
(Reporting by Agnieszka Flak; Editing by Keith Weir and Alexandra Hudson)
By Agnieszka Flak